Theorem opabbidv 4178

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 4177	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  {copab 4172

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 4174

This theorem is referenced by:  opabbii  4179  csbopabg  4190  xpeq1  4765  xpeq2  4766  opabbi2dv  4906  csbcnvg  4941  resopab2  5087  mptcnv  5167  cores  5268  xpcom  5311  dffn5im  5724  f1oiso2  6002  f1ocnvd  6259  ofreq  6272  f1od2  6433  shftfvalg  11511  shftfval  11514  2shfti  11524  prdsex  13503  prdsval  13507  releqgg  13958  eqgex  13959  eqgfval  13960  dvdsrvald  14260  dvdsrpropdg  14314  aprval  14451  aprap  14458  aprprop  14461  lmfval  15107  lgsquadlem3  16001  wksfval  16366  trlsfvalg  16427  eupthsg  16489