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Theorem rexeqbi1dv 2816
 Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
Hypothesis
Ref Expression
raleqd.1 (A = B → (φψ))
Assertion
Ref Expression
rexeqbi1dv (A = B → (x A φx B ψ))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem rexeqbi1dv
StepHypRef Expression
1 rexeq 2808 . 2 (A = B → (x A φx B φ))
2 raleqd.1 . . 3 (A = B → (φψ))
32rexbidv 2635 . 2 (A = B → (x B φx B ψ))
41, 3bitrd 244 1 (A = B → (x A φx B ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∃wrex 2615 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  frd  5922
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