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Theorem raleqbi1dv 2487
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
raleqd.1 (A = B → (φψ))
Assertion
Ref Expression
raleqbi1dv (A = B → (x A φx B ψ))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem raleqbi1dv
StepHypRef Expression
1 raleq 2479 . 2 (A = B → (x A φx B φ))
2 raleqd.1 . . 3 (A = B → (φψ))
32ralbidv 2300 . 2 (A = B → (x B φx B ψ))
41, 3bitrd 177 1 (A = B → (x A φx B ψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1226  wral 2280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285
This theorem is referenced by:  peano5  4244  isoeq4  5365  bj-indeq  7291  peano5set  7301  bj-nntrans  7312
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