ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexeq GIF version

Theorem rexeq 2563
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 2228 . 2 𝑥𝐴
2 nfcv 2228 . 2 𝑥𝐵
31, 2rexeqf 2559 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wrex 2360
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365
This theorem is referenced by:  rexeqi  2567  rexeqdv  2569  rexeqbi1dv  2571  unieq  3662  bnd2  4008  exss  4054  qseq1  6340  finexdc  6618  supeq1  6681  isomni  6792  bj-nn0sucALT  11873  strcoll2  11878  sscoll2  11883
  Copyright terms: Public domain W3C validator