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Theorem oprabbidv 7215
Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.)
Hypothesis
Ref Expression
oprabbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
oprabbidv (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
Distinct variable groups:   𝑥,𝑧,𝜑   𝑦,𝑧,𝜑
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem oprabbidv
StepHypRef Expression
1 nfv 1908 . 2 𝑥𝜑
2 nfv 1908 . 2 𝑦𝜑
3 nfv 1908 . 2 𝑧𝜑
4 oprabbidv.1 . 2 (𝜑 → (𝜓𝜒))
51, 2, 3, 4oprabbid 7214 1 (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  {coprab 7152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-9 2116  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-oprab 7155
This theorem is referenced by:  oprabbii  7216  mpoeq123dva  7223  mpoeq3dva  7226  resoprab2  7264  erovlem  8386  joinfval  17603  meetfval  17617  odumeet  17742  odujoin  17744  mppsval  32704  csbmpo123  34482  unceq  34737  uncf  34739  unccur  34743
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