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Theorem rexeq 2551
 Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
Assertion
Ref Expression
rexeq (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexeq
StepHypRef Expression
1 nfcv 2220 . 2 𝑥𝐴
2 nfcv 2220 . 2 𝑥𝐵
31, 2rexeqf 2547 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103   = wceq 1285  ∃wrex 2350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355 This theorem is referenced by:  rexeqi  2555  rexeqdv  2557  rexeqbi1dv  2559  unieq  3618  bnd2  3955  exss  3990  qseq1  6220  supeq1  6458  bj-nn0sucALT  10931  strcoll2  10936  sscoll2  10941
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