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Theorem opabbii 3880
Description: Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
Hypothesis
Ref Expression
opabbii.1 (𝜑𝜓)
Assertion
Ref Expression
opabbii {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Proof of Theorem opabbii
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqid 2085 . 2 𝑧 = 𝑧
2 opabbii.1 . . . 4 (𝜑𝜓)
32a1i 9 . . 3 (𝑧 = 𝑧 → (𝜑𝜓))
43opabbidv 3879 . 2 (𝑧 = 𝑧 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
51, 4ax-mp 7 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1287  {copab 3873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-opab 3875
This theorem is referenced by:  mptv  3909  fconstmpt  4452  xpundi  4461  xpundir  4462  inxp  4537  cnvco  4588  resopab  4722  opabresid  4729  cnvi  4799  cnvun  4800  cnvin  4802  cnvxp  4813  cnvcnv3  4843  coundi  4895  coundir  4896  mptun  5106  fvopab6  5353  cbvoprab1  5671  cbvoprab12  5673  dmoprabss  5681  mpt2mptx  5690  resoprab  5692  ov6g  5733  dfoprab3s  5911  dfoprab3  5912  dfoprab4  5913  mapsncnv  6398  xpcomco  6488  dmaddpq  6875  dmmulpq  6876  recmulnqg  6887  enq0enq  6927  ltrelxr  7484  ltxr  9171  shftidt2  10155
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