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Theorem opabbii 3995
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 2139 . 2 𝑧 = 𝑧
2 opabbii.1 . . . 4 (𝜑𝜓)
32a1i 9 . . 3 (𝑧 = 𝑧 → (𝜑𝜓))
43opabbidv 3994 . 2 (𝑧 = 𝑧 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
51, 4ax-mp 5 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1331  {copab 3988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-opab 3990
This theorem is referenced by:  mptv  4025  fconstmpt  4586  xpundi  4595  xpundir  4596  inxp  4673  cnvco  4724  resopab  4863  opabresid  4872  cnvi  4943  cnvun  4944  cnvin  4946  cnvxp  4957  cnvcnv3  4988  coundi  5040  coundir  5041  mptun  5254  fvopab6  5517  cbvoprab1  5843  cbvoprab12  5845  dmoprabss  5853  mpomptx  5862  resoprab  5867  ov6g  5908  dfoprab3s  6088  dfoprab3  6089  dfoprab4  6090  mapsncnv  6589  xpcomco  6720  dmaddpq  7201  dmmulpq  7202  recmulnqg  7213  enq0enq  7253  ltrelxr  7839  ltxr  9576  shftidt2  10618  lmfval  12377  lmbr  12398  cnmptid  12466
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