Theorem opabbidv 4197

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)

Hypothesis

Ref	Expression
opabbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

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

Step	Hyp	Ref	Expression
1		nfv 1581	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1581	. 2 ⊢ Ⅎ𝑦𝜑
3		opabbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	1, 2, 3	opabbid 4196	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   = wceq 1402  {copab 4191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-opab 4193

This theorem is used by:  opabbii  4198  csbopabg  4209  xpeq1  4788  xpeq2  4789  opabbi2dv  4929  csbcnvg  4964  resopab2  5110  mptcnv  5190  cores  5291  xpcom  5334  dffn5im  5748  f1oiso2  6033  f1ocnvd  6292  f1o3d  6298  ofreq  6306  f1od2  6471  shftfvalg  11585  shftfval  11588  2shfti  11598  releqgg  14025  eqgex  14026  eqgfval  14027  prdsex  14174  prdsval  14175  dvdsrvald  14402  dvdsrpropdg  14456  aprval  14593  aprap  14600  aprprop  14603  lmfval  15296  lgsquadlem3  16210  wksfval  16575  trlsfvalg  16636  eupthsg  16698