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Theorem rexeqbidva 3150
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 3045 . 2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
3 raleqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
43rexeqdv 3140 . 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 384   = wceq 1481  wcel 1988  wrex 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915
This theorem is referenced by:  catpropd  16350  istrkgb  25335  istrkgcb  25336  istrkge  25337  isperp  25588  perpcom  25589  eengtrkg  25846  eengtrkge  25847  afsval  30723  matunitlindflem2  33377
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