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Theorem rexeqdv 2690
Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
Hypothesis
Ref Expression
raleq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
rexeqdv (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rexeqdv
StepHypRef Expression
1 raleq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 rexeq 2684 . 2 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
31, 2syl 14 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1363  wrex 2466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471
This theorem is referenced by:  rexeqbidv  2696  rexeqbidva  2698  fnunirn  5781  cbvexfo  5800  fival  6982  nninfwlpoimlemg  7186  nninfwlpoimlemginf  7187  nninfwlpoim  7189  genipv  7521  exfzdc  10253  zproddc  11600  infssuzex  11963  nninfdcex  11967  ennnfonelemrnh  12430  grppropd  12912  dvdsrpropdg  13378  cnpfval  13922
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