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Theorem oprabbii 5923
Description: Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
oprabbii.1 (𝜑𝜓)
Assertion
Ref Expression
oprabbii {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem oprabbii
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . 2 𝑤 = 𝑤
2 oprabbii.1 . . . 4 (𝜑𝜓)
32a1i 9 . . 3 (𝑤 = 𝑤 → (𝜑𝜓))
43oprabbidv 5922 . 2 (𝑤 = 𝑤 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
51, 4ax-mp 5 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  {coprab 5869
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-oprab 5872
This theorem is referenced by:  oprab4  5939  mpov  5958  dfxp3  6188  tposmpo  6275  oviec  6634  dfplpq2  7331  dfmpq2  7332  dfmq0qs  7406  dfplq0qs  7407  addsrpr  7722  mulsrpr  7723  addcnsr  7811  mulcnsr  7812  addvalex  7821
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