ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvopab1v GIF version

Theorem cbvopab1v 3914
Description: Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
Hypothesis
Ref Expression
cbvopab1v.1 (𝑥 = 𝑧 → (𝜑𝜓))
Assertion
Ref Expression
cbvopab1v {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧   𝜑,𝑧   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦,𝑧)

Proof of Theorem cbvopab1v
StepHypRef Expression
1 nfv 1466 . 2 𝑧𝜑
2 nfv 1466 . 2 𝑥𝜓
3 cbvopab1v.1 . 2 (𝑥 = 𝑧 → (𝜑𝜓))
41, 2, 3cbvopab1 3911 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  {copab 3898
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-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator