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

Theorem cbvrexv 2551
Description: Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
Hypothesis
Ref Expression
cbvralv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrexv (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvrexv
StepHypRef Expression
1 nfv 1437 . 2 𝑦𝜑
2 nfv 1437 . 2 𝑥𝜓
3 cbvralv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrex 2547 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wrex 2324
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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329
This theorem is referenced by:  cbvrex2v  2559  reu7  2759  reusv3  4220  funcnvuni  4996  fun11iun  5175  fvelimab  5257  fliftfun  5464  grpridd  5725  frecsuc  6022  nnaordex  6131  supmoti  6399  cardval3ex  6423  prarloclemlo  6650  prarloclem3  6653  cauappcvgprlemdisj  6807  cauappcvgprlemladdru  6812  cauappcvgprlemladdrl  6813  cauappcvgpr  6818  caucvgprlemdisj  6830  caucvgprlemcl  6832  caucvgprlemladdfu  6833  caucvgprlemladdrl  6834  caucvgpr  6838  caucvgprprlemell  6841  caucvgprprlemelu  6842  caucvgprprlemlol  6854  caucvgprprlemclphr  6861  caucvgprprlemexbt  6862  nntopi  7026  axcaucvglemres  7031  ublbneg  8645  qbtwnzlemstep  9205  qbtwnzlemshrink  9206  rebtwn2zlemstep  9209  rebtwn2zlemshrink  9210  cvg1nlemres  9812  resqrexlemoverl  9848  resqrexlemsqa  9851  resqrexlemex  9852  odd2np1lem  10183  divalglemeunn  10233  divalglemeuneg  10235  bj-nn0sucALT  10490
  Copyright terms: Public domain W3C validator