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Theorem mo4f 2626
 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 ⊢ Ⅎ𝑦𝜑 with essentially the same proof. (Contributed by NM, 10-Apr-2004.) Remove dependency on ax-13 2379. (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 1915 . . 3 𝑦𝜑
21mo3 2623 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
3 mo4f.1 . . . . . 6 𝑥𝜓
4 mo4f.2 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbiev 2322 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65anbi2i 625 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓))
76imbi1i 353 . . 3 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝑦))
872albii 1822 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
92, 8bitri 278 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  Ⅎwnf 1785  [wsb 2069  ∃*wmo 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598 This theorem is referenced by:  mo4OLD  2627  axextmo  2774  mob2  3655  moop2  5360
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