Theorem opabbii 4151

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 2229	. 2 ⊢ 𝑧 = 𝑧
2		opabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	opabbidv 4150	. 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 1395  {copab 4144

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-opab 4146

This theorem is referenced by:  mptv  4181  fconstmpt  4766  xpundi  4775  xpundir  4776  inxp  4856  cnvco  4907  resopab  5049  opabresid  5058  cnvi  5133  cnvun  5134  cnvin  5136  cnvxp  5147  cnvcnv3  5178  coundi  5230  coundir  5231  mptun  5455  fvopab6  5733  cbvoprab1  6082  cbvoprab12  6084  dmoprabss  6092  mpomptx  6101  resoprab  6106  ov6g  6149  dfoprab3s  6342  dfoprab3  6343  dfoprab4  6344  mapsncnv  6850  xpcomco  6993  dmaddpq  7577  dmmulpq  7578  recmulnqg  7589  enq0enq  7629  ltrelxr  8218  ltxr  9983  shftidt2  11358  prdsex  13317  prdsval  13321  prdsbaslemss  13322  releqgg  13772  eqgex  13773  dvdsrzring  14582  lmfval  14882  lmbr  14902  cnmptid  14970  lgsquadlem3  15773  wksfval  16063