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Theorem List for Metamath Proof Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2sp 2201 A double specialization (see sp 2198). Another double specialization, closer to PM*11.1, is 2stdpc4 2489. (Contributed by BJ, 15-Sep-2018.)
(∀𝑥𝑦𝜑𝜑)

Theoremspsd 2202 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))

Theorem19.2g 2203 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 2056 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
(∀𝑥𝜑 → ∃𝑦𝜑)

Theorem19.21bi 2204 Inference form of 19.21 2220 and also deduction form of sp 2198. (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜓)       (𝜑𝜓)

Theorem19.21bbi 2205 Inference removing double quantifier. Version of 19.21bi 2204 with two quanditiers. (Contributed by NM, 20-Apr-1994.)
(𝜑 → ∀𝑥𝑦𝜓)       (𝜑𝜓)

Theorem19.23bi 2206 Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2225. (Contributed by NM, 12-Mar-1993.)
(∃𝑥𝜑𝜓)       (𝜑𝜓)

Theoremnexr 2207 Inference associated with the contrapositive of 19.8a 2197. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ ∃𝑥𝜑        ¬ 𝜑

Theoremqexmid 2208 Quantified excluded middle (see exmid 430). Also known as the drinker paradox (if 𝜑(𝑥) is interpreted as "𝑥 drinks", then this theorem tells that there exists a person such that, if this person drinks, then everyone drinks). Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)
𝑥(𝜑 → ∀𝑥𝜑)

Theoremnf5r 2209 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1857 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Theoremnf5ri 2210 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       (𝜑 → ∀𝑥𝜑)

Theoremnf5rd 2211 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))

Theoremnfim1 2212 A closed form of nfim 1972. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1857 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theoremnfan1 2213 A closed form of nfan 1975. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1857 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theorem19.3 2214 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. See 19.3v 2061 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑       (∀𝑥𝜑𝜑)

Theorem19.9d 2215 A deduction version of one direction of 19.9 2217. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf 1857 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(𝜓 → Ⅎ𝑥𝜑)       (𝜓 → (∃𝑥𝜑𝜑))

Theorem19.9t 2216 A closed version of 19.9 2217. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
(Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))

Theorem19.9 2217 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 2060 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
𝑥𝜑       (∃𝑥𝜑𝜑)

Theorem19.21t 2218 Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2220. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1857 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theorem19.21tOLDOLD 2219 Obsolete proof of 19.21t 2218 as of 3-Nov-2021. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1857 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theorem19.21 2220 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See 19.21v 2015 for a version requiring fewer axioms. See also 19.21h 2266. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1857 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theoremstdpc5 2221 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis 𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 2093. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 2014 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2166. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theoremstdpc5OLD 2222 Obsolete proof of stdpc5 2221 as of 11-Oct-2021. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2166. (Revised by Wolf Lammen, 4-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theorem19.21-2 2223 Version of 19.21 2220 with two quantifiers. (Contributed by NM, 4-Feb-2005.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))

Theorem19.23t 2224 Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2225. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1857 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theorem19.23 2225 Theorem 19.23 of [Margaris] p. 90. See 19.23v 2018 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theoremalimd 2226 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1885. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalrimi 2227 Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2220. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremalrimdd 2228 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2220. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremalrimd 2229 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2220. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   𝑥𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremeximd 2230 Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1908. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremexlimi 2231 Inference associated with 19.23 2225. See exlimiv 2005 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜓    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theoremexlimd 2232 Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

Theoremexlimdd 2233 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theoremnexd 2234 Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theoremalbid 2235 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbid 2236 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremnfbidf 2237 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1857 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Theorem19.16 2238 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∀𝑥𝜓))

Theorem19.17 2239 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorem19.27 2240 Theorem 19.27 of [Margaris] p. 90. See 19.27v 2071 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28 2241 Theorem 19.28 of [Margaris] p. 90. See 19.28v 2072 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theorem19.19 2242 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Theorem19.36 2243 Theorem 19.36 of [Margaris] p. 90. See 19.36v 2067 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.36i 2244 Inference associated with 19.36 2243. See 19.36iv 2021 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
𝑥𝜓    &   𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorem19.37 2245 Theorem 19.37 of [Margaris] p. 90. See 19.37v 2073 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Theorem19.32 2246 Theorem 19.32 of [Margaris] p. 90. See 19.32v 2016 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

Theorem19.31 2247 Theorem 19.31 of [Margaris] p. 90. See 19.31v 2017 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.41 2248 Theorem 19.41 of [Margaris] p. 90. See 19.41v 2024 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.42-1 2249 One direction of 19.42 2250. (Contributed by Wolf Lammen, 10-Jul-2021.)
𝑥𝜑       ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.42 2250 Theorem 19.42 of [Margaris] p. 90. See 19.42v 2028 for a version requiring fewer axioms. See exan 1935 for an immediate version. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theorem19.44 2251 Theorem 19.44 of [Margaris] p. 90. See 19.44v 2075 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.45 2252 Theorem 19.45 of [Margaris] p. 90. See 19.45v 2076 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Theoremequsalv 2253* Version of equsal 2434 with a dv condition, which does not require ax-13 2389. See equsalvw 2084 for a version with two dv conditions requiring fewer axioms. See also the dual form equsexv 2254. (Contributed by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexv 2254* Version of equsex 2435 with a dv condition, which does not require ax-13 2389. See equsexvw 2085 for a version with two dv conditions requiring fewer axioms. See also the dual form equsalv 2253. (Contributed by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremsbequ1 2255 An equality theorem for substitution. (Contributed by NM, 16-May-1993.)
(𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))

Theoremsbequ12 2256 An equality theorem for substitution. (Contributed by NM, 14-May-1993.)
(𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))

Theoremsbequ12r 2257 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))

Theoremsbequ12a 2258 An equality theorem for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Jun-2019.)
(𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))

Theoremsbid 2259 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)
([𝑥 / 𝑥]𝜑𝜑)

Theoremspimv1 2260* Version of spim 2397 with a dv condition, which does not require ax-13 2389. See spimvw 2080 for a version with two dv conditions, requiring fewer axioms, and spimv 2400 for another variant. (Contributed by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremnf5 2261 Alternate definition of df-nf 1857. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1857 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Theoremnf6 2262 An alternate definition of df-nf 1857. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))

Theoremnf5d 2263 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremnf5di 2264 Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either 𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.)
(𝜑 → Ⅎ𝑥𝜑)       𝑥𝜑

Theorem19.9h 2265 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
(𝜑 → ∀𝑥𝜑)       (∃𝑥𝜑𝜑)

Theorem19.21h 2266 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See also 19.21 2220 and 19.21v 2015. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜑 → ∀𝑥𝜑)       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theorem19.23h 2267 Theorem 19.23 of [Margaris] p. 90. See 19.23 2225. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜓 → ∀𝑥𝜓)       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theoremequsalhw 2268* Weaker version of equsalh 2437 with a dv condition which does not require ax-13 2389. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexhv 2269* Version of equsexh 2438 with a dv condition, which does not require ax-13 2389. (Contributed by BJ, 31-May-2019.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremhbim1 2270 A closed form of hbim 2272. (Contributed by NM, 2-Jun-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhbimd 2271 Deduction form of bound-variable hypothesis builder hbim 2272. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))

Theoremhbim 2272 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhban 2273 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhb3an 2274 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))

Theoremaxc4 2275 Show that the original axiom ax-c4 34671 can be derived from ax-4 1884 (alim 1885), ax-10 2166 (hbn1 2167), sp 2198 and propositional calculus. See ax4fromc4 34681 for the rederivation of ax-4 1884 from ax-c4 34671.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

(∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremaxc4i 2276 Inference version of axc4 2275. (Contributed by NM, 3-Jan-1993.)
(∀𝑥𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theoremaxc7 2277 Show that the original axiom ax-c7 34672 can be derived from ax-10 2166 (hbn1 2167) , sp 2198 and propositional calculus. See ax10fromc7 34682 for the rederivation of ax-10 2166 from ax-c7 34672.

Normally, axc7 2277 should be used rather than ax-c7 34672, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc7e 2278 Abbreviated version of axc7 2277 using the existential quantifier. (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝑥𝜑𝜑)

Theoremaxc16g 2279* Generalization of axc16 2280. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 2088 }, theorem ax12v 2195. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2389, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2284. (Revised by Wolf Lammen, 11-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theoremaxc16 2280* Proof of older axiom ax-c16 34679. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremaxc16gb 2281* Biconditional strengthening of axc16g 2279. (Contributed by NM, 15-May-1993.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))

Theoremaxc16nf 2282* If dtru 5004 is false, then there is only one element in the universe, so everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2181. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2166. (Revised by Wolf lammen, 12-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Theoremaxc11v 2283* Version of axc11 2454 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2088 }, from ax12v 2195 (contrary to axc11 2454 which seems to require the full ax-12 2194 and ax-13 2389). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremaxc11rv 2284* Version of axc11r 2330 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2088 }, from ax12v 2195 (contrary to axc11 2454 which seems to require the full ax-12 2194). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))

Theoremaxc11rvOLD 2285* Obsolete proof of axc11rv 2284 as of 11-Oct-2021. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))

Theoremaxc11vOLD 2286* Obsolete proof of axc11v 2283 as of 11-Oct-2021. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremmodal-b 2287 The analogue in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
(𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Theorem19.9ht 2288 A closed version of 19.9h 2265. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Theoremhbnt 2289 Closed theorem version of bound-variable hypothesis builder hbn 2291. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) (Proof shortened by Wolf Lammen, 14-Oct-2021.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

TheoremhbntOLD 2290 Obsolete proof of hbnt 2289 as of 13-Oct-2021. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Theoremhbn 2291 If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.)
(𝜑 → ∀𝑥𝜑)       𝜑 → ∀𝑥 ¬ 𝜑)

Theoremhbnd 2292 Deduction form of bound-variable hypothesis builder hbn 2291. (Contributed by NM, 3-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))

Theoremexlimih 2293 Inference associated with 19.23 2225. See exlimiv 2005 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜓 → ∀𝑥𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theoremexlimdh 2294 Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-Jan-1997.)
(𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

Theoremsb56 2295* Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 2045. The implication "to the left" is equs4 2433 and does not require any dv condition (but the version with a dv condition, equs4v 2083, requires fewer axioms). Theorem equs45f 2485 replaces the dv condition with a non-freeness hypothesis and equs5 2486 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2254 in place of equsex 2435 in order to remove dependency on ax-13 2389. (Revised by BJ, 20-Dec-2020.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremhba1 2296 The setvar 𝑥 is not free in 𝑥𝜑. This corresponds to the axiom (4) of modal logic. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 12-Oct-2021.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)

TheoremhbexOLD 2297 Obsolete proof of hbex 2301 as of 16-Oct-2021. (Contributed by NM, 12-Mar-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∃𝑦𝜑 → ∀𝑥𝑦𝜑)

Theoremnfal 2298 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥𝑦𝜑

Theoremnfex 2299 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Reduce symbol count in nfex 2299, hbex 2301. (Revised by Wolf Lammen, 16-Oct-2021.)
𝑥𝜑       𝑥𝑦𝜑

TheoremnfexOLD 2300 Obsolete proof of nfex 2299 as of 16-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       𝑥𝑦𝜑

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