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Theorem rexeqbii 3264
Description: Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
rexeqbii.1 𝐴 = 𝐵
rexeqbii.2 (𝜓𝜒)
Assertion
Ref Expression
rexeqbii (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)

Proof of Theorem rexeqbii
StepHypRef Expression
1 rexeqbii.1 . . . 4 𝐴 = 𝐵
21eleq2i 2898 . . 3 (𝑥𝐴𝑥𝐵)
3 rexeqbii.2 . . 3 (𝜓𝜒)
42, 3anbi12i 620 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54rexbii2 3249 1 (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1656  wcel 2164  wrex 3118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-cleq 2818  df-clel 2821  df-rex 3123
This theorem is referenced by:  bnj882  31538
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