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Theorem opabbii 3851
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 2056 . 2 𝑧 = 𝑧
2 opabbii.1 . . . 4 (𝜑𝜓)
32a1i 9 . . 3 (𝑧 = 𝑧 → (𝜑𝜓))
43opabbidv 3850 . 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 102   = wceq 1259  {copab 3844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-opab 3846
This theorem is referenced by:  mptv  3880  fconstmpt  4414  xpundi  4423  xpundir  4424  inxp  4497  cnvco  4547  resopab  4679  opabresid  4686  cnvi  4755  cnvun  4756  cnvin  4758  cnvxp  4769  cnvcnv3  4797  coundi  4849  coundir  4850  mptun  5056  fvopab6  5291  cbvoprab1  5603  cbvoprab12  5605  dmoprabss  5613  mpt2mptx  5622  resoprab  5624  ov6g  5665  dfoprab3s  5843  dfoprab3  5844  dfoprab4  5845  xpcomco  6330  dmaddpq  6534  dmmulpq  6535  recmulnqg  6546  enq0enq  6586  ltrelxr  7138  ltxr  8795  shftidt2  9660
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