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Theorem rexeqbidva 3333
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 3177 . 2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
3 raleqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
43rexeqdv 3327 . 2 (𝜑 → (∃𝑥𝐴 𝜒 ↔ ∃𝑥𝐵 𝜒))
52, 4bitrd 279 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-rex 3071
This theorem is referenced by:  rexeqbidvv  3336  catpropd  17752  addsval  27995  istrkgcb  28464  isperp  28720  perpcom  28721  eengtrkg  29001  eengtrkge  29002  opprqusdrng  33521  fldextrspunlsplem  33723  afsval  34686  matunitlindflem2  37624  rrxlines  48654
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