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Theorem raleqdv 2813
 Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
Hypothesis
Ref Expression
raleq1d.1 (φA = B)
Assertion
Ref Expression
raleqdv (φ → (x A ψx B ψ))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem raleqdv
StepHypRef Expression
1 raleq1d.1 . 2 (φA = B)
2 raleq 2807 . 2 (A = B → (x A ψx B ψ))
31, 2syl 15 1 (φ → (x A ψx B ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∀wral 2614 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-ral 2619 This theorem is referenced by:  raleqbidv  2819  raleqbidva  2821
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