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Theorem mo4f 2562
Description: At-most-one quantifier expressed using implicit substitution. Note that the disjoint variable condition on 𝑦, 𝜑 can be replaced by the nonfreeness hypothesis 𝑦𝜑 with essentially the same proof. (Contributed by NM, 10-Apr-2004.) Remove dependency on ax-13 2372. (Revised by Wolf Lammen, 19-Jan-2023.)
Hypotheses
Ref Expression
mo4f.1 𝑥𝜓
mo4f.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
mo4f (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem mo4f
StepHypRef Expression
1 nfv 1918 . . 3 𝑦𝜑
21mo3 2559 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
3 mo4f.1 . . . . . 6 𝑥𝜓
4 mo4f.2 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbiev 2309 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65anbi2i 624 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓))
76imbi1i 350 . . 3 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝑦))
872albii 1823 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
92, 8bitri 275 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wnf 1786  [wsb 2068  ∃*wmo 2533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535
This theorem is referenced by:  axextmo  2708  mob2  3712  moop2  5503
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