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Theorem mo4fOLD 2584
 Description: Obsolete version of mo4f 2583 as of 19-Jan-2023. (Contributed by NM, 10-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
mo4f.1 𝑥𝜓
mo4f.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
mo4fOLD (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem mo4fOLD
StepHypRef Expression
1 nfv 1957 . . 3 𝑦𝜑
21mo3OLD 2581 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
3 mo4f.1 . . . . . 6 𝑥𝜓
4 mo4f.2 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2484 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65anbi2i 616 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓))
76imbi1i 341 . . 3 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝑦))
872albii 1864 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
92, 8bitri 267 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599  Ⅎwnf 1827  [wsb 2011  ∃*wmo 2549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551 This theorem is referenced by: (None)
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