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Theorem opabbidv 3989
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)
Hypothesis
Ref Expression
opabbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
opabbidv (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem opabbidv
StepHypRef Expression
1 nfv 1508 . 2 𝑥𝜑
2 nfv 1508 . 2 𝑦𝜑
3 opabbidv.1 . 2 (𝜑 → (𝜓𝜒))
41, 2, 3opabbid 3988 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  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-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-opab 3985
This theorem is referenced by:  opabbii  3990  csbopabg  4001  xpeq1  4548  xpeq2  4549  opabbi2dv  4683  csbcnvg  4718  resopab2  4861  mptcnv  4936  cores  5037  xpcom  5080  dffn5im  5460  f1oiso2  5721  f1ocnvd  5965  ofreq  5978  f1od2  6125  shftfvalg  10583  shftfval  10586  2shfti  10596  lmfval  12350
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