Theorem opabbidv 4126

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 1552	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1552	. 2 ⊢ Ⅎ𝑦𝜑
3		opabbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	1, 2, 3	opabbid 4125	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373  {copab 4120

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-opab 4122

This theorem is referenced by:  opabbii  4127  csbopabg  4138  xpeq1  4707  xpeq2  4708  opabbi2dv  4845  csbcnvg  4880  resopab2  5025  mptcnv  5104  cores  5205  xpcom  5248  dffn5im  5647  f1oiso2  5919  f1ocnvd  6171  ofreq  6185  f1od2  6344  shftfvalg  11244  shftfval  11247  2shfti  11257  prdsex  13216  prdsval  13220  releqgg  13671  eqgex  13672  eqgfval  13673  reldvdsrsrg  13969  dvdsrvald  13970  dvdsrpropdg  14024  aprval  14159  aprap  14163  lmfval  14779  lgsquadlem3  15671