Theorem opabbidv 4099

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 4098	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  {copab 4093

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 4095

This theorem is referenced by:  opabbii  4100  csbopabg  4111  xpeq1  4677  xpeq2  4678  opabbi2dv  4815  csbcnvg  4850  resopab2  4993  mptcnv  5072  cores  5173  xpcom  5216  dffn5im  5606  f1oiso2  5874  f1ocnvd  6125  ofreq  6139  f1od2  6293  shftfvalg  10983  shftfval  10986  2shfti  10996  prdsex  12940  releqgg  13350  eqgex  13351  eqgfval  13352  reldvdsrsrg  13648  dvdsrvald  13649  dvdsrpropdg  13703  aprval  13838  aprap  13842  lmfval  14428  lgsquadlem3  15320