Theorem opabbii 4085

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 2189	. 2 ⊢ 𝑧 = 𝑧
2		opabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	opabbidv 4084	. 2 ⊢ (𝑧 = 𝑧 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
5	1, 4	ax-mp 5	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Syntax hints:   ↔ wb 105   = wceq 1364  {copab 4078

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-opab 4080

This theorem is referenced by:  mptv  4115  fconstmpt  4688  xpundi  4697  xpundir  4698  inxp  4776  cnvco  4827  resopab  4966  opabresid  4975  cnvi  5048  cnvun  5049  cnvin  5051  cnvxp  5062  cnvcnv3  5093  coundi  5145  coundir  5146  mptun  5362  fvopab6  5628  cbvoprab1  5963  cbvoprab12  5965  dmoprabss  5973  mpomptx  5982  resoprab  5987  ov6g  6029  dfoprab3s  6209  dfoprab3  6210  dfoprab4  6211  mapsncnv  6713  xpcomco  6844  dmaddpq  7396  dmmulpq  7397  recmulnqg  7408  enq0enq  7448  ltrelxr  8036  ltxr  9793  shftidt2  10859  prdsex  12740  releqgg  13125  eqgex  13126  dvdsrzring  13863  lmfval  14089  lmbr  14110  cnmptid  14178