Theorem opabbidv 4181

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 1577	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1577	. 2 ⊢ Ⅎ𝑦𝜑
3		opabbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	1, 2, 3	opabbid 4180	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  {copab 4175

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-opab 4177

This theorem is referenced by:  opabbii  4182  csbopabg  4193  xpeq1  4768  xpeq2  4769  opabbi2dv  4909  csbcnvg  4944  resopab2  5090  mptcnv  5170  cores  5271  xpcom  5314  dffn5im  5727  f1oiso2  6006  f1ocnvd  6265  f1o3d  6271  ofreq  6279  f1od2  6444  shftfvalg  11531  shftfval  11534  2shfti  11544  releqgg  13976  eqgex  13977  eqgfval  13978  prdsex  14117  prdsval  14118  dvdsrvald  14341  dvdsrpropdg  14395  aprval  14532  aprap  14539  aprprop  14542  lmfval  15187  lgsquadlem3  16081  wksfval  16446  trlsfvalg  16507  eupthsg  16569