Theorem opabbidv 4100

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 1542	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1542	. 2 ⊢ Ⅎ𝑦𝜑
3		opabbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	1, 2, 3	opabbid 4099	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  {copab 4094

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-opab 4096

This theorem is referenced by:  opabbii  4101  csbopabg  4112  xpeq1  4678  xpeq2  4679  opabbi2dv  4816  csbcnvg  4851  resopab2  4994  mptcnv  5073  cores  5174  xpcom  5217  dffn5im  5609  f1oiso2  5877  f1ocnvd  6129  ofreq  6143  f1od2  6302  shftfvalg  11000  shftfval  11003  2shfti  11013  prdsex  12971  prdsval  12975  releqgg  13426  eqgex  13427  eqgfval  13428  reldvdsrsrg  13724  dvdsrvald  13725  dvdsrpropdg  13779  aprval  13914  aprap  13918  lmfval  14512  lgsquadlem3  15404