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Theorem raleq 2479
Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
Assertion
Ref Expression
raleq (A = B → (x A φx B φ))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem raleq
StepHypRef Expression
1 nfcv 2156 . 2 xA
2 nfcv 2156 . 2 xB
31, 2raleqf 2475 1 (A = B → (x A φx B φ))
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:  raleqi  2483  raleqdv  2485  raleqbi1dv  2487  sbralie  2520  inteq  3588  iineq1  3641  bnd2  3896  ordeq  4054  fncnv  4887  funimaexglem  4904  isoeq4  5365  acexmidlemv  5430  tfrlem1  5841  tfrlemisucaccv  5856  tfrlemi1  5863  tfrlemi14d  5864  tfrlemi14  5865  tfrexlem  5866  setindis  7381  bdsetindis  7383  strcoll2  7397
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