Theorem opabbii 3824

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 2040	. 2 ⊢ 𝑧 = 𝑧
2		opabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	opabbidv 3823	. 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 98   = wceq 1243  {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-opab 3819

This theorem is referenced by:  mptv  3853  fconstmpt  4387  xpundi  4396  xpundir  4397  inxp  4470  cnvco  4520  resopab  4652  opabresid  4659  cnvi  4728  cnvun  4729  cnvin  4731  cnvxp  4742  cnvcnv3  4770  coundi  4822  coundir  4823  mptun  5029  fvopab6  5264  cbvoprab1  5576  cbvoprab12  5578  dmoprabss  5586  mpt2mptx  5595  resoprab  5597  ov6g  5638  dfoprab3s  5816  dfoprab3  5817  dfoprab4  5818  xpcomco  6300  dmaddpq  6475  dmmulpq  6476  recmulnqg  6487  enq0enq  6527  ltrelxr  7078  ltxr  8693  shftidt2  9407