Theorem opabbii 4150

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 4149	. 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 4143

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 4145

This theorem is referenced by:  mptv  4180  fconstmpt  4765  xpundi  4774  xpundir  4775  inxp  4855  cnvco  4906  resopab  5048  opabresid  5057  cnvi  5132  cnvun  5133  cnvin  5135  cnvxp  5146  cnvcnv3  5177  coundi  5229  coundir  5230  mptun  5454  fvopab6  5730  cbvoprab1  6075  cbvoprab12  6077  dmoprabss  6085  mpomptx  6094  resoprab  6099  ov6g  6142  dfoprab3s  6334  dfoprab3  6335  dfoprab4  6336  mapsncnv  6840  xpcomco  6981  dmaddpq  7562  dmmulpq  7563  recmulnqg  7574  enq0enq  7614  ltrelxr  8203  ltxr  9967  shftidt2  11338  prdsex  13297  prdsval  13301  prdsbaslemss  13302  releqgg  13752  eqgex  13753  dvdsrzring  14561  lmfval  14860  lmbr  14881  cnmptid  14949  lgsquadlem3  15752  wksfval  16028