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Theorem rexeqbidv 2820
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
Hypotheses
Ref Expression
raleqbidv.1 (φA = B)
raleqbidv.2 (φ → (ψχ))
Assertion
Ref Expression
rexeqbidv (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 rexeqbidv
StepHypRef Expression
1 raleqbidv.1 . . 3 (φA = B)
21rexeqdv 2814 . 2 (φ → (x A ψx B ψ))
3 raleqbidv.2 . . 3 (φ → (ψχ))
43rexbidv 2635 . 2 (φ → (x B ψx B χ))
52, 4bitrd 244 1 (φ → (x A ψx B χ))
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:  elsuci  4414  nnsucelr  4428  clos1basesucg  5884  el2c  6191  dflec3  6221  nmembers1lem3  6270
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