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Theorem mo3 2506
Description: Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
Hypothesis
Ref Expression
mo3.1 𝑦𝜑
Assertion
Ref Expression
mo3 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfmo1 2480 . . 3 𝑥∃*𝑥𝜑
2 mo3.1 . . . . 5 𝑦𝜑
32nfmo 2486 . . . 4 𝑦∃*𝑥𝜑
4 mo2v 2476 . . . . 5 (∃*𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
5 sp 2052 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = 𝑧))
6 spsbim 2393 . . . . . . . . 9 (∀𝑥(𝜑𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝑥 = 𝑧))
7 equsb3 2431 . . . . . . . . 9 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
86, 7syl6ib 241 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
95, 8anim12d 586 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑧𝑦 = 𝑧)))
10 equtr2 1953 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
119, 10syl6 35 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1211exlimiv 1857 . . . . 5 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
134, 12sylbi 207 . . . 4 (∃*𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
143, 13alrimi 2081 . . 3 (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
151, 14alrimi 2081 . 2 (∃*𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
16 nfs1v 2436 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜑
17 pm3.21 464 . . . . . . . . 9 ([𝑦 / 𝑥]𝜑 → (𝜑 → (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1817imim1d 82 . . . . . . . 8 ([𝑦 / 𝑥]𝜑 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑𝑥 = 𝑦)))
1916, 18alimd 2080 . . . . . . 7 ([𝑦 / 𝑥]𝜑 → (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
2019com12 32 . . . . . 6 (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
2120aleximi 1758 . . . . 5 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
222sb8e 2424 . . . . 5 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
232mo2 2478 . . . . 5 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2421, 22, 233imtr4g 285 . . . 4 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃*𝑥𝜑))
25 moabs 2500 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2624, 25sylibr 224 . . 3 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
2726alcoms 2034 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
2815, 27impbii 199 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480  wex 1703  wnf 1707  [wsb 1879  ∃*wmo 2470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474
This theorem is referenced by:  mo  2507  eu2  2508  mo4f  2515  2mo  2550  rmo3  3526  isarep2  5976  mo5f  29308  rmo3f  29319  rmo4fOLD  29320  bnj580  30968  pm14.12  38448
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