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Theorem opabbidv 4043
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 1515 . 2 𝑥𝜑
2 nfv 1515 . 2 𝑦𝜑
3 opabbidv.1 . 2 (𝜑 → (𝜓𝜒))
41, 2, 3opabbid 4042 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1342  {copab 4037
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-opab 4039
This theorem is referenced by:  opabbii  4044  csbopabg  4055  xpeq1  4613  xpeq2  4614  opabbi2dv  4748  csbcnvg  4783  resopab2  4926  mptcnv  5001  cores  5102  xpcom  5145  dffn5im  5527  f1oiso2  5790  f1ocnvd  6035  ofreq  6048  f1od2  6195  shftfvalg  10750  shftfval  10753  2shfti  10763  lmfval  12759
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