Theorem opabbii 3990

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 2137	. 2 ⊢ 𝑧 = 𝑧
2		opabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	opabbidv 3989	. 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 1331  {copab 3983

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-opab 3985

This theorem is referenced by:  mptv  4020  fconstmpt  4581  xpundi  4590  xpundir  4591  inxp  4668  cnvco  4719  resopab  4858  opabresid  4867  cnvi  4938  cnvun  4939  cnvin  4941  cnvxp  4952  cnvcnv3  4983  coundi  5035  coundir  5036  mptun  5249  fvopab6  5510  cbvoprab1  5836  cbvoprab12  5838  dmoprabss  5846  mpomptx  5855  resoprab  5860  ov6g  5901  dfoprab3s  6081  dfoprab3  6082  dfoprab4  6083  mapsncnv  6582  xpcomco  6713  dmaddpq  7180  dmmulpq  7181  recmulnqg  7192  enq0enq  7232  ltrelxr  7818  ltxr  9555  shftidt2  10597  lmfval  12350  lmbr  12371  cnmptid  12439