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Theorem raleqbidv 2819
 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
raleqbidv (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 raleqbidv
StepHypRef Expression
1 raleqbidv.1 . . 3 (φA = B)
21raleqdv 2813 . 2 (φ → (x A ψx B ψ))
3 raleqbidv.2 . . 3 (φ → (ψχ))
43ralbidv 2634 . 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  ∀wral 2614 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-ral 2619 This theorem is referenced by:  fmpt2x  5730
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