Theorem opabbidv 4175

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 4174	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  {copab 4169

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-opab 4171

This theorem is referenced by:  opabbii  4176  csbopabg  4187  xpeq1  4762  xpeq2  4763  opabbi2dv  4903  csbcnvg  4938  resopab2  5084  mptcnv  5164  cores  5265  xpcom  5308  dffn5im  5721  f1oiso2  5999  f1ocnvd  6256  ofreq  6269  f1od2  6430  shftfvalg  11496  shftfval  11499  2shfti  11509  prdsex  13471  prdsval  13475  releqgg  13926  eqgex  13927  eqgfval  13928  dvdsrvald  14227  dvdsrpropdg  14281  aprval  14417  aprap  14421  lmfval  15045  lgsquadlem3  15939  wksfval  16304  trlsfvalg  16365  eupthsg  16427