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Theorem rexeqbid 3128
 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 3112 . . 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 3033 . 2 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑥𝐵 𝜒))
95, 8bitrd 267 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  Ⅎwnf 1699  Ⅎwnfc 2738  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902 This theorem is referenced by:  iuneq12df  4480
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