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Theorem cbvopab1v 3999
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 1508 . 2 𝑧𝜑
2 nfv 1508 . 2 𝑥𝜓
3 cbvopab1v.1 . 2 (𝑥 = 𝑧 → (𝜑𝜓))
41, 2, 3cbvopab1 3996 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  {copab 3983
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 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985
This theorem is referenced by: (None)
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