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Theorem rexeqbidva 3292
Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
Hypotheses
Ref Expression
raleqbidva.1 (𝜑𝐴 = 𝐵)
raleqbidva.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rexeqbidva (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rexeqbidva
StepHypRef Expression
1 raleqbidva.2 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21rexbidva 3185 . 2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
3 raleqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
43rexeqdv 3282 . 2 (𝜑 → (∃𝑥𝐴 𝜒 ↔ ∃𝑥𝐵 𝜒))
52, 4bitrd 268 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wrex 3049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rex 3054
This theorem is referenced by:  catpropd  16568  istrkgb  25551  istrkgcb  25552  istrkge  25553  isperp  25804  perpcom  25805  eengtrkg  26062  eengtrkge  26063  afsval  31056  matunitlindflem2  33717
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