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Theorem oprabbidv 6875
Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (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 1992 . 2 𝑥𝜑
2 nfv 1992 . 2 𝑦𝜑
3 nfv 1992 . 2 𝑧𝜑
4 oprabbidv.1 . 2 (𝜑 → (𝜓𝜒))
51, 2, 3, 4oprabbid 6874 1 (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  {coprab 6815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-oprab 6818
This theorem is referenced by:  oprabbii  6876  mpt2eq123dva  6882  mpt2eq3dva  6885  resoprab2  6923  erovlem  8012  joinfval  17222  meetfval  17236  odumeet  17361  odujoin  17363  mppsval  31797  csbmpt22g  33506  unceq  33717  uncf  33719  unccur  33723
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