Theorem opabbii 3935

Description: Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)

Hypothesis

Ref	Expression
opabbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
opabbii	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Proof of Theorem opabbii

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2100	. 2 ⊢ 𝑧 = 𝑧
2		opabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	opabbidv 3934	. 2 ⊢ (𝑧 = 𝑧 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
5	1, 4	ax-mp 7	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Syntax hints:   ↔ wb 104   = wceq 1299  {copab 3928

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-opab 3930

This theorem is referenced by:  mptv  3965  fconstmpt  4524  xpundi  4533  xpundir  4534  inxp  4611  cnvco  4662  resopab  4799  opabresid  4808  cnvi  4879  cnvun  4880  cnvin  4882  cnvxp  4893  cnvcnv3  4924  coundi  4976  coundir  4977  mptun  5190  fvopab6  5449  cbvoprab1  5775  cbvoprab12  5777  dmoprabss  5785  mpomptx  5794  resoprab  5799  ov6g  5840  dfoprab3s  6018  dfoprab3  6019  dfoprab4  6020  mapsncnv  6519  xpcomco  6649  dmaddpq  7088  dmmulpq  7089  recmulnqg  7100  enq0enq  7140  ltrelxr  7697  ltxr  9403  shftidt2  10445  lmfval  12143  lmbr  12163  cnmptid  12231