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Theorem rexeq 2521
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 2192 . 2 𝑥𝐴
2 nfcv 2192 . 2 𝑥𝐵
31, 2rexeqf 2517 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1257  wrex 2322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327
This theorem is referenced by:  rexeqi  2525  rexeqdv  2527  rexeqbi1dv  2529  unieq  3614  bnd2  3951  exss  3988  qseq1  6182  bj-nn0sucALT  10433  strcoll2  10438  sscoll2  10443
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