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Theorem oprabbii 7479
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 2731 . 2 𝑤 = 𝑤
2 oprabbii.1 . . . 4 (𝜑𝜓)
32a1i 11 . . 3 (𝑤 = 𝑤 → (𝜑𝜓))
43oprabbidv 7478 . 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 setvar class
Syntax hints:  wb 205   = wceq 1540  {coprab 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-oprab 7416
This theorem is referenced by:  oprab4  7498  mpov  7523  dfxp3  8051  tposmpo  8252  addsrpr  11074  mulsrpr  11075  addcnsr  11134  mulcnsr  11135  joinfval2  18332  meetfval2  18346  dfxrn2  37550
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