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Theorem raleq 2499
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 2175 . 2 xA
2 nfcv 2175 . 2 xB
31, 2raleqf 2495 1 (A = B → (x A φx B φ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wral 2300
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305
This theorem is referenced by:  raleqi  2503  raleqdv  2505  raleqbi1dv  2507  sbralie  2540  inteq  3609  iineq1  3662  bnd2  3917  ordeq  4075  fncnv  4908  funimaexglem  4925  isoeq4  5387  acexmidlemv  5453  tfrlem1  5864  tfr0  5878  tfrlemisucaccv  5880  tfrlemi1  5887  tfrlemi14d  5888  tfrexlem  5889  setindis  9397  bdsetindis  9399  strcoll2  9413
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