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Theorem List for Metamath Proof Explorer - 33401-33500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwl-impchain-mp-1 33401 This theorem is in fact a copy of wl-syl 33376, and repeated here to demonstrate a recursive proof scheme. The number '1' in the theorem name indicates that a chain of length 1 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜒𝜓)    &   (𝜓𝜑)       (𝜒𝜑)

Theoremwl-impchain-mp-2 33402 This theorem is in fact a copy of wl-syl6 33384, and repeated here to demonstrate a recursive proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜃 → (𝜒𝜓))    &   (𝜓𝜑)       (𝜃 → (𝜒𝜑))

Theoremwl-impchain-com-1.x 33403 It is often convenient to have the antecedent under focus in first position, so we can apply immediate theorem forms (as opposed to deduction, tautology form). This series of theorems swaps the first with the last antecedent in an implication chain. This kind of swapping is self-inverse, whence we prefer it over, say, rotating theorems. A consequent can hide a tail of a longer chain, so theorems of this series appear as swapping a pair of antecedents with fixed offsets. This form of swapping antecedents is flexible enough to allow for any permutation of antecedents in an implication chain.

The first elements of this series correspond to com12 32, com13 88, com14 96 and com15 101 in the main part.

The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-mp-x 33399 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.)

Theoremwl-impchain-com-1.1 33404 A degenerate form of antecedent swapping. The number '1' in the theorem name indicates that it handles a chain of length 1.

Since there is just one antecedent in the chain, there is nothing to swap. Non-degenerated forms begin with wl-impchain-com-1.2 33405, for more see there. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜓𝜑)       (𝜓𝜑)

Theoremwl-impchain-com-1.2 33405 This theorem is in fact a copy of wl-com12 33388, and repeated here to demonstrate a simple proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified.

See wl-impchain-com-1.x 33403 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜒 → (𝜓𝜑))       (𝜓 → (𝜒𝜑))

Theoremwl-impchain-com-1.3 33406 This theorem is in fact a copy of com13 88, and repeated here to demonstrate a simple proof scheme. The number '3' in the theorem name indicates that a chain of length 3 is modified.

See wl-impchain-com-1.x 33403 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜃 → (𝜒 → (𝜓𝜑)))       (𝜓 → (𝜒 → (𝜃𝜑)))

Theoremwl-impchain-com-1.4 33407 This theorem is in fact a copy of com14 96, and repeated here to demonstrate a simple proof scheme. The number '4' in the theorem name indicates that a chain of length 4 is modified.

See wl-impchain-com-1.x 33403 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜂 → (𝜃 → (𝜒 → (𝜓𝜑))))       (𝜓 → (𝜃 → (𝜒 → (𝜂𝜑))))

Theoremwl-impchain-com-n.m 33408 This series of theorems allow swapping any two antecedents in an implication chain. The theorem names follow a pattern wl-impchain-com-n.m with integral numbers n < m, that swaps the m-th antecedent with n-th one in an implication chain. It is sufficient to restrict the length of the chain to m, too, since the consequent can be assumed to be the tail right of the m-th antecedent of any arbitrary sized implication chain. We further assume n > 1, since the wl-impchain-com-1.x 33403 series already covers the special case n = 1.

Being able to swap any two antecedents in an implication chain lays the foundation of permuting its antecedents arbitrarily.

The proofs of this series aim at automated proofing using a simple scheme. Any instance of this series is a triple step of swapping the first and n-th antecedent, then the first and the m-th, then the first and the n-th antecedent again. Each of these steps is an instance of the wl-impchain-com-1.x 33403 series. (Contributed by Wolf Lammen, 17-Nov-2019.)

Theoremwl-impchain-com-2.3 33409 This theorem is in fact a copy of com23 86. It starts a series of theorems named after wl-impchain-com-n.m 33408. For more information see there. (Contributed by Wolf Lammen, 12-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜃 → (𝜒 → (𝜓𝜑)))       (𝜃 → (𝜓 → (𝜒𝜑)))

Theoremwl-impchain-com-2.4 33410 This theorem is in fact a copy of com24 95. It is another instantiation of theorems named after wl-impchain-com-n.m 33408. For more information see there. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜂 → (𝜃 → (𝜒 → (𝜓𝜑))))       (𝜂 → (𝜓 → (𝜒 → (𝜃𝜑))))

Theoremwl-impchain-com-3.2.1 33411 This theorem is in fact a copy of com3r 87. The proof is an example of how to arrive at arbitrary permutations of antecedents, using only swapping theorems. The recursion principle is to first swap the correct antecedent to the position just before the consequent, and then employ a theorem handling an implication chain of length one less to reorder the others. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 → (𝜒 → (𝜓𝜑)))       (𝜓 → (𝜃 → (𝜒𝜑)))

Theoremwl-impchain-a1-x 33412 If an implication chain is assumed (hypothesis) or proven (theorem) to hold, then we may add any extra antecedent to it, without changing its truth. This is expressed in its simplest form in wl-a1i 33381, that allows us prepending an arbitrary antecedent to an implication chain. Using our antecedent swapping theorems described in wl-impchain-com-n.m 33408, we may then move such a prepended antecedent to any desired location within all antecedents. The first series of theorems of this kind adds a single antecedent somewhere to an implication chain. The appended number in the theorem name indicates its position within all antecedents, 1 denoting the head position. A second theorem series extends this idea to multiple additions (TODO).

Adding antecedents to an implication chain usually weakens their universality. The consequent afterwards dependends on more conditions than before, which renders the implication chain less versatile. So you find this proof technique mostly when you adjust a chain to a hypothesis of a rule. A common case are syllogisms merging two implication chains into one.

The first elements of the first series correspond to a1i 11, a1d 25 and a1dd 50 in the main part.

The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-com-1.x 33403 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.)

Theoremwl-impchain-a1-1 33413 Inference rule, a copy of a1i 11. Head start of a recursive proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       (𝜓𝜑)

Theoremwl-impchain-a1-2 33414 Inference rule, a copy of a1d 25. First recursive proof based on the previous instance. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))

Theoremwl-impchain-a1-3 33415 Inference rule, a copy of a1dd 50. A recursive proof depending on previous instances, and demonstrating the proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜃𝜒)))

20.17.3  An alternative axiom ~ ax-13

Axiomax-wl-13v 33416* A version of ax13v 2283 with a distinctor instead of a distinct variable expression.

Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1879. So essentially this theorem states, that a distinct variable condition between set variables can be replaced with a distinctor expression. (Contributed by Wolf Lammen, 23-Jul-2021.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremwl-ax13lem1 33417* A version of ax-wl-13v 33416 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

20.17.4  Other stuff

Theoremwl-jarri 33418 Dropping a nested antecedent. This theorem is one of two reversions of ja 173. Since ja 173 is reversible, a nested (chain of) implication(s) is just a packed notation of two or more theorems/hypotheses with a common consequent. axc5c7 34515 is an instance of this idea. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theoremwl-jarli 33419 Dropping a nested consequent. This theorem is one of two reversions of ja 173. Since ja 173 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. axc5c7 34515 is an instance of this idea. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       𝜑𝜒)

Theoremwl-mps 33420 Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremwl-syls1 33421 Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremwl-syls2 33422 Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   ((𝜑𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)

Theoremwl-embant 33423 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)

Theoremwl-orel12 33424 In a conjunctive normal form a pair of nodes like (𝜑𝜓) ∧ (¬ 𝜑𝜒) eliminates the need of a node (𝜓𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.)
(((𝜑𝜓) ∧ (¬ 𝜑𝜒)) → (𝜓𝜒))

Theoremwl-cases2-dnf 33425 A particular instance of orddi 931 and anddi 932 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1016, and is related to consensus 1023. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1033 and dfifp4 1036, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)
(((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))

Theoremwl-dfnan2 33426 An alternative definition of "nand" based on imnan 437. See df-nan 1488 for the original definition. This theorem allows various shortenings. (Contributed by Wolf Lammen, 26-Jun-2020.)
((𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))

Theoremwl-nancom 33427 The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Revised by Wolf Lammen, 26-Jun-2020.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theoremwl-nannan 33428 Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.)
((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Theoremwl-nannot 33429 Show equivalence between negation and the Nicod version. To derive nic-dfneg 1635, apply nanbi 1494. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.)
𝜑 ↔ (𝜑𝜑))

Theoremwl-nanbi1 33430 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (Revised by Wolf Lammen, 27-Jun-2020.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremwl-nanbi2 33431 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (Revised by Wolf Lammen, 27-Jun-2020.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremwl-naev 33432* If some set variables can assume different values, then any two distinct set variables cannot always be the same. (Contributed by Wolf Lammen, 10-Aug-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣)

Theoremwl-hbae1 33433 This specialization of hbae 2348 does not depend on ax-11 2074. (Contributed by Wolf Lammen, 8-Aug-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)

Theoremwl-naevhba1v 33434* An instance of hbn1w 2015 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-hbnaev 33435* Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. The latter condition can actually be lifted, but this version is easier to prove. The proof does not use ax-10 2059. (Contributed by Wolf Lammen, 9-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-spae 33436 Prove an instance of sp 2091 from ax-13 2282 and Tarski's FOL only, without distinct variable conditions. The antecedent 𝑥𝑥 = 𝑦 holds in a multi-object universe only if 𝑦 is substituted for 𝑥, or vice versa, i.e. both variables are effectively the same. The converse ¬ ∀𝑥𝑥 = 𝑦 indicates that both variables are distinct, and it so provides a simple translation of a distinct variable condition to a logical term. In case studies 𝑥𝑥 = 𝑦 and ¬ ∀𝑥𝑥 = 𝑦 can help eliminating distinct variable conditions.

The antecedent 𝑥𝑥 = 𝑦 is expressed in the theorem's name by the abbreviation ae standing for 'all equal'.

Note that we cannot provide a logical predicate telling us directly whether a logical expression contains a particular variable, as such a construct would usually contradict ax-12 2087.

Note that this theorem is also provable from ax-12 2087 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 1954 and spaev 2020 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.)

(∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)

Theoremwl-cbv3vv 33437* Avoiding ax-11 2074. (Contributed by Wolf Lammen, 30-Aug-2021.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremwl-speqv 33438* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2091 is provable from Tarski's FOL and ax13v 2283 only. Note that this reverts the implication in ax13lem1 2284, so in fact 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦𝑧 = 𝑦))

Theoremwl-19.8eqv 33439* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2090 is provable from Tarski's FOL and ax13v 2283 only. Note that this reverts the implication in ax13lem2 2332, so in fact 𝑥 = 𝑦 → (∃𝑥𝑧 = 𝑦𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦))

Theoremwl-19.2reqv 33440* Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1949 is provable from Tarski's FOL and ax13v 2283 only. Note that in conjunction with 19.2 1949 in fact 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremwl-dveeq12 33441* The current form of ax-13 2282 has a particular disadvantage: The condition ¬ 𝑥 = 𝑦 is less versatile than the general form ¬ ∀𝑥𝑥 = 𝑦. You need ax-10 2059 to arrive at the more general form presented here. You need 19.8a 2090 (or ax-12 2087) to restore 𝑦 = 𝑧 from 𝑥𝑦 = 𝑧 again. (Contributed by Wolf Lammen, 9-Jun-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremwl-nfalv 33442* If 𝑥 is not present in 𝜑, it is not free in 𝑦𝜑. (Contributed by Wolf Lammen, 11-Jan-2020.)
𝑥𝑦𝜑

Theoremwl-nfimf1 33443 An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1865 in dvelimdf 2366 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)
(∀𝑥𝜑 → (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥𝜓))

Theoremwl-nfnbi 33444 Being free does not depend on an outside negation in an expression. This theorem is slightly more general than nfn 1824 or nfnd 1825. (Contributed by Wolf Lammen, 5-May-2018.)
(Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Theoremwl-nfae1 33445 Unlike nfae 2349, this specialized theorem avoids ax-11 2074. (Contributed by Wolf Lammen, 26-Jun-2019.)
𝑥𝑦 𝑦 = 𝑥

Theoremwl-nfnae1 33446 Unlike nfnae 2351, this specialized theorem avoids ax-11 2074. (Contributed by Wolf Lammen, 27-Jun-2019.)
𝑥 ¬ ∀𝑦 𝑦 = 𝑥

Theoremwl-aetr 33447 A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))

Theoremwl-dral1d 33448 A version of dral1 2356 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, 𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 33459 and nf5di 2157 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))

Theoremwl-cbvalnaed 33449 wl-cbvalnae 33450 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))    &   (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremwl-cbvalnae 33450 A more general version of cbval 2307 when non-free properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2365, nfsb2 2388 or dveeq1 2336. (Contributed by Wolf Lammen, 4-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)    &   (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremwl-exeq 33451 The semantics of 𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
(∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))

Theoremwl-aleq 33452 The semantics of 𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
(∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))

Theoremwl-nfeqfb 33453 Extend nfeqf 2337 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.)
(Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))

Theoremwl-nfs1t 33454 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2393. (Contributed by Wolf Lammen, 27-Jul-2019.)
(Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theoremwl-equsald 33455 Deduction version of equsal 2327. (Contributed by Wolf Lammen, 27-Jul-2019.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))

Theoremwl-equsal 33456 A useful equivalence related to substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) It seems proving wl-equsald 33455 first, and then deriving more specialized versions wl-equsal 33456 and wl-equsal1t 33457 then is more efficient than the other way round, which is possible, too. See also equsal 2327. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremwl-equsal1t 33457 The expression 𝑥 = 𝑦 in antecedent position plays an important role in predicate logic, namely in implicit substitution. However, occasionally it is irrelevant, and can safely be dropped. A sufficient condition for this is when 𝑥 (or 𝑦 or both) is not free in 𝜑.

This theorem is more fundamental than equsal 2327, spimt 2289 or sbft 2407, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)

(Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))

Theoremwl-equsalcom 33458 This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑦 = 𝑥𝜑))

Theoremwl-equsal1i 33459 The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.)
(Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)    &   (𝑥 = 𝑦𝜑)       𝜑

Theoremwl-sb6rft 33460 A specialization of wl-equsal1t 33457. Closed form of sb6rf 2451. (Contributed by Wolf Lammen, 27-Jul-2019.)
(Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))

Theoremwl-sbrimt 33461 Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2424. (Contributed by Wolf Lammen, 26-Jul-2019.)
(Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))

Theoremwl-sblimt 33462 Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2424. (Contributed by Wolf Lammen, 26-Jul-2019.)
(Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))

Theoremwl-sb8t 33463 Substitution of variable in universal quantifier. Closed form of sb8 2452. (Contributed by Wolf Lammen, 27-Jul-2019.)
(∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

Theoremwl-sb8et 33464 Substitution of variable in universal quantifier. Closed form of sb8e 2453. (Contributed by Wolf Lammen, 27-Jul-2019.)
(∀𝑥𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))

Theoremwl-sbhbt 33465 Closed form of sbhb 2466. Characterizing the expression 𝜑 → ∀𝑥𝜑 using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.)
(∀𝑥𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Theoremwl-sbnf1 33466 Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2467. Note: This theorem shows that sbnf2 2467 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)
(∀𝑥𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Theoremwl-equsb3 33467 equsb3 2460 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)
(¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))

Theoremwl-equsb4 33468 Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))

Theoremwl-sb6nae 33469 Version of sb6 2457 suitable for elimination of unnecessary dv restrictions. (Contributed by Wolf Lammen, 28-Jul-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremwl-sb5nae 33470 Version of sb5 2458 suitable for elimination of unnecessary dv restrictions. (Contributed by Wolf Lammen, 28-Jul-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))

Theoremwl-2sb6d 33471 Version of 2sb6 2472 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
(𝜑 → ¬ ∀𝑦 𝑦 = 𝑥)    &   (𝜑 → ¬ ∀𝑦 𝑦 = 𝑤)    &   (𝜑 → ¬ ∀𝑦 𝑦 = 𝑧)    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)       (𝜑 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))

Theoremwl-sbcom2d-lem1 33472* Lemma used to prove wl-sbcom2d 33474. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
((𝑢 = 𝑦𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))

Theoremwl-sbcom2d-lem2 33473* Lemma used to prove wl-sbcom2d 33474. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
(¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) → 𝜑)))

Theoremwl-sbcom2d 33474 Version of sbcom2 2473 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
(𝜑 → ¬ ∀𝑥 𝑥 = 𝑤)    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)    &   (𝜑 → ¬ ∀𝑧 𝑧 = 𝑦)       (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))

Theoremwl-sbalnae 33475 A theorem used in elimination of disjoint variable restrictions by replacing them with distinctors. (Contributed by Wolf Lammen, 25-Jul-2019.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremwl-sbal1 33476* A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) Proof is based on wl-sbalnae 33475 now. See also sbal1 2488. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremwl-sbal2 33477* Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002.) Proof is based on wl-sbalnae 33475 now. See also sbal2 2489. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremwl-lem-exsb 33478* This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremwl-lem-nexmo 33479 This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
(¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑧))

Theoremwl-lem-moexsb 33480* The antecedent 𝑥(𝜑𝑥 = 𝑧) relates to ∃*𝑥𝜑, but is better suited for usage in proofs. Note that no distinct variable restriction is placed on 𝜑.

This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)

(∀𝑥(𝜑𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))

Theoremwl-alanbii 33481 This theorem extends alanimi 1784 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.)
(𝜑 ↔ (𝜓𝜒))       (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))

Theoremwl-mo2df 33482 Version of mo2 2507 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate dv conditions. (Contributed by Wolf Lammen, 11-Aug-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   (𝜑 → Ⅎ𝑦𝜓)       (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))

Theoremwl-mo2tf 33483 Closed form of mo2 2507 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 20-Sep-2020.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremwl-eudf 33484 Version of df-eu 2502 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate dv conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   (𝜑 → Ⅎ𝑦𝜓)       (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))

Theoremwl-eutf 33485 Closed form of df-eu 2502 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremwl-euequ1f 33486 euequ1 2504 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)

Theoremwl-mo2t 33487* Closed form of mo2 2507. (Contributed by Wolf Lammen, 18-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremwl-mo3t 33488* Closed form of mo3 2536. (Contributed by Wolf Lammen, 18-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Theoremwl-sb8eut 33489 Substitution of variable in universal quantifier. Closed form of sb8eu 2532. (Contributed by Wolf Lammen, 11-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))

Theoremwl-sb8mot 33490 Substitution of variable in universal quantifier. Closed form of sb8mo 2533.

This theorem relates to wl-mo3t 33488, since replacing 𝜑 with [𝑦 / 𝑥]𝜑 in the latter yields subexpressions like [𝑥 / 𝑦][𝑦 / 𝑥]𝜑, which can be reduced to 𝜑 via sbft 2407 and sbco 2440. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 33488 in a simple fashion, unfortunately only if 𝑥 and 𝑦 are known to be distinct. To avoid any hassle with distinctors, we prefer to derive this theorem independently, ignoring the close connection between both theorems. From an educational standpoint, one would assume wl-mo3t 33488 to be more fundamental, as it hints how the "at most one" objects on both sides of the biconditional correlate (they are the same), if they exist at all, and then prove this theorem from it. (Contributed by Wolf Lammen, 11-Aug-2019.)

(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))

Axiomax-wl-11v 33491* Version of ax-11 2074 with distinct variable conditions. Currently implemented as an axiom to detect unintended references to the foundational axiom ax-11 2074. It will later be converted into a theorem directly based on ax-11 2074. (Contributed by Wolf Lammen, 28-Jun-2019.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremwl-ax11-lem1 33492 A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑦 𝑦 = 𝑧))

Theoremwl-ax11-lem2 33493* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)

Theoremwl-ax11-lem3 33494* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)

Theoremwl-ax11-lem4 33495* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-ax11-lem5 33496 Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))

Theoremwl-ax11-lem6 33497* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))

Theoremwl-ax11-lem7 33498 Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥(¬ ∀𝑥 𝑥 = 𝑦𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))

Theoremwl-ax11-lem8 33499* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑))

Theoremwl-ax11-lem9 33500 The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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