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Theorem rexeqbidva 2822
 Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
Hypotheses
Ref Expression
raleqbidva.1 (φA = B)
raleqbidva.2 ((φ x A) → (ψχ))
Assertion
Ref Expression
rexeqbidva (φ → (x A ψx B 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 ((φ x A) → (ψχ))
21rexbidva 2631 . 2 (φ → (x A ψx A χ))
3 raleqbidva.1 . . 3 (φA = B)
43rexeqdv 2814 . 2 (φ → (x A χx B χ))
52, 4bitrd 244 1 (φ → (x A ψx B χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620 This theorem is referenced by: (None)
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