Theorem opabbii 4003

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 2140	. 2 ⊢ 𝑧 = 𝑧
2		opabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	opabbidv 4002	. 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 104   = wceq 1332  {copab 3996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-opab 3998

This theorem is referenced by:  mptv  4033  fconstmpt  4594  xpundi  4603  xpundir  4604  inxp  4681  cnvco  4732  resopab  4871  opabresid  4880  cnvi  4951  cnvun  4952  cnvin  4954  cnvxp  4965  cnvcnv3  4996  coundi  5048  coundir  5049  mptun  5262  fvopab6  5525  cbvoprab1  5851  cbvoprab12  5853  dmoprabss  5861  mpomptx  5870  resoprab  5875  ov6g  5916  dfoprab3s  6096  dfoprab3  6097  dfoprab4  6098  mapsncnv  6597  xpcomco  6728  dmaddpq  7211  dmmulpq  7212  recmulnqg  7223  enq0enq  7263  ltrelxr  7849  ltxr  9592  shftidt2  10636  lmfval  12400  lmbr  12421  cnmptid  12489