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Theorem mo3 2494
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 2468 . . 3 𝑥∃*𝑥𝜑
2 mo3.1 . . . . 5 𝑦𝜑
32nfmo 2474 . . . 4 𝑦∃*𝑥𝜑
4 mo2v 2464 . . . . 5 (∃*𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
5 sp 2040 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = 𝑧))
6 spsbim 2381 . . . . . . . . 9 (∀𝑥(𝜑𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝑥 = 𝑧))
7 equsb3 2419 . . . . . . . . 9 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
86, 7syl6ib 239 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
95, 8anim12d 583 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑧𝑦 = 𝑧)))
10 equtr2 1940 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
119, 10syl6 34 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1211exlimiv 1844 . . . . 5 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
134, 12sylbi 205 . . . 4 (∃*𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
143, 13alrimi 2068 . . 3 (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
151, 14alrimi 2068 . 2 (∃*𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
16 nfs1v 2424 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜑
17 pm3.21 462 . . . . . . . . 9 ([𝑦 / 𝑥]𝜑 → (𝜑 → (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
1817imim1d 79 . . . . . . . 8 ([𝑦 / 𝑥]𝜑 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑𝑥 = 𝑦)))
1916, 18alimd 2067 . . . . . . 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 1748 . . . . 5 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
222sb8e 2412 . . . . 5 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
232mo2 2466 . . . . 5 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2421, 22, 233imtr4g 283 . . . 4 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃*𝑥𝜑))
25 moabs 2488 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2624, 25sylibr 222 . . 3 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
2726alcoms 2021 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
2815, 27impbii 197 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wex 1694  wnf 1698  [wsb 1866  ∃*wmo 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462
This theorem is referenced by:  mo  2495  eu2  2496  mo4f  2503  2mo  2538  rmo3  3493  isarep2  5878  mo5f  28542  rmo3f  28553  rmo4fOLD  28554  bnj580  30071  pm14.12  37468
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