Theorem opabbidv 4195

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 4194	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  {copab 4189

This theorem was proved from 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 theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-opab 4191

This theorem is referenced by:  opabbii  4196  csbopabg  4207  xpeq1  4786  xpeq2  4787  opabbi2dv  4927  csbcnvg  4962  resopab2  5108  mptcnv  5188  cores  5289  xpcom  5332  dffn5im  5745  f1oiso2  6027  f1ocnvd  6286  f1o3d  6292  ofreq  6300  f1od2  6465  shftfvalg  11566  shftfval  11569  2shfti  11579  releqgg  14006  eqgex  14007  eqgfval  14008  prdsex  14155  prdsval  14156  dvdsrvald  14383  dvdsrpropdg  14437  aprval  14574  aprap  14581  aprprop  14584  lmfval  15277  lgsquadlem3  16181  wksfval  16546  trlsfvalg  16607  eupthsg  16669