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Theorem cbvoprab12v 7495
Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)
Hypothesis
Ref Expression
cbvoprab12v.1 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))
Assertion
Ref Expression
cbvoprab12v {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣   𝜑,𝑤,𝑣   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑧,𝑤,𝑣)

Proof of Theorem cbvoprab12v
StepHypRef Expression
1 nfv 1909 . 2 𝑤𝜑
2 nfv 1909 . 2 𝑣𝜑
3 nfv 1909 . 2 𝑥𝜓
4 nfv 1909 . 2 𝑦𝜓
5 cbvoprab12v.1 . 2 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))
61, 2, 3, 4, 5cbvoprab12 7494 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  {coprab 7406
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-oprab 7409
This theorem is referenced by:  cpnnen  16179
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