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Theorem rexeqbid 3372
Description: Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
raleqbid.0 𝑥𝜑
raleqbid.1 𝑥𝐴
raleqbid.2 𝑥𝐵
raleqbid.3 (𝜑𝐴 = 𝐵)
raleqbid.4 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rexeqbid (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Proof of Theorem rexeqbid
StepHypRef Expression
1 raleqbid.3 . . 3 (𝜑𝐴 = 𝐵)
2 raleqbid.1 . . . 4 𝑥𝐴
3 raleqbid.2 . . . 4 𝑥𝐵
42, 3rexeqf 3354 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
51, 4syl 17 . 2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
6 raleqbid.0 . . 3 𝑥𝜑
7 raleqbid.4 . . 3 (𝜑 → (𝜓𝜒))
86, 7rexbid 3282 . 2 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑥𝐵 𝜒))
95, 8bitrd 282 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wnf 1785  wnfc 2939  wrex 3110
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-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rex 3115
This theorem is referenced by:  iuneq12df  4910
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