Theorem opabbidv 4153

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 1574	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1574	. 2 ⊢ Ⅎ𝑦𝜑
3		opabbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	1, 2, 3	opabbid 4152	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  {copab 4147

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-opab 4149

This theorem is referenced by:  opabbii  4154  csbopabg  4165  xpeq1  4737  xpeq2  4738  opabbi2dv  4877  csbcnvg  4912  resopab2  5058  mptcnv  5137  cores  5238  xpcom  5281  dffn5im  5687  f1oiso2  5963  f1ocnvd  6220  ofreq  6234  f1od2  6395  shftfvalg  11372  shftfval  11375  2shfti  11385  prdsex  13345  prdsval  13349  releqgg  13800  eqgex  13801  eqgfval  13802  dvdsrvald  14100  dvdsrpropdg  14154  aprval  14289  aprap  14293  lmfval  14910  lgsquadlem3  15801  wksfval  16133  trlsfvalg  16192  eupthsg  16254