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Theorem rexeqbii 3285
 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 2881 . . 3 (𝑥𝐴𝑥𝐵)
3 rexeqbii.2 . . 3 (𝜓𝜒)
42, 3anbi12i 629 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54rexbii2 3208 1 (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-rex 3112 This theorem is referenced by:  bnj882  32308  satfbrsuc  32726
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