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Theorem rexeq 2500
 Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
Assertion
Ref Expression
rexeq (A = B → (x A φx B φ))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem rexeq
StepHypRef Expression
1 nfcv 2175 . 2 xA
2 nfcv 2175 . 2 xB
31, 2rexeqf 2496 1 (A = B → (x A φx B φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃wrex 2301 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-bndl 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-rex 2306 This theorem is referenced by:  rexeqi  2504  rexeqdv  2506  rexeqbi1dv  2508  unieq  3580  bnd2  3917  exss  3954  qseq1  6090  bj-nn0sucALT  9438  strcoll2  9443  sscoll2  9448
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