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Theorem rexeqi 2631
Description: Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
raleq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
rexeqi (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexeqi
StepHypRef Expression
1 raleq1i.1 . 2 𝐴 = 𝐵
2 rexeq 2627 . 2 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
31, 2ax-mp 5 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)
Colors of variables: wff set class
Syntax hints:  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:  rexrab2  2851  rexprg  3575  rextpg  3577  rexxp  4683  rexrnmpo  5886  0ct  6992  arch  8981  infssuzex  11649  gcdsupex  11653  gcdsupcl  11654  txbas  12437
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