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Theorem rexeq 2627
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 2281 . 2 𝑥𝐴
2 nfcv 2281 . 2 𝑥𝐵
31, 2rexeqf 2623 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wrex 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422
This theorem is referenced by:  rexeqi  2631  rexeqdv  2633  rexeqbi1dv  2635  unieq  3745  bnd2  4097  exss  4149  qseq1  6477  finexdc  6796  supeq1  6873  isomni  7008  ismkv  7027  sup3exmid  8722  exmidunben  11946  neifval  12319  cnprcl2k  12385  bj-nn0sucALT  13206  strcoll2  13211  sscoll2  13216
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