Theorem opabbii 4119

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 2206	. 2 ⊢ 𝑧 = 𝑧
2		opabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	opabbidv 4118	. 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 1373  {copab 4112

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-opab 4114

This theorem is referenced by:  mptv  4149  fconstmpt  4730  xpundi  4739  xpundir  4740  inxp  4820  cnvco  4871  resopab  5012  opabresid  5021  cnvi  5096  cnvun  5097  cnvin  5099  cnvxp  5110  cnvcnv3  5141  coundi  5193  coundir  5194  mptun  5417  fvopab6  5689  cbvoprab1  6030  cbvoprab12  6032  dmoprabss  6040  mpomptx  6049  resoprab  6054  ov6g  6097  dfoprab3s  6289  dfoprab3  6290  dfoprab4  6291  mapsncnv  6795  xpcomco  6936  dmaddpq  7512  dmmulpq  7513  recmulnqg  7524  enq0enq  7564  ltrelxr  8153  ltxr  9917  shftidt2  11218  prdsex  13176  prdsval  13180  prdsbaslemss  13181  releqgg  13631  eqgex  13632  dvdsrzring  14440  lmfval  14739  lmbr  14760  cnmptid  14828  lgsquadlem3  15631