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Theorem cbvoprab12v 5579
 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 1421 . 2 𝑤𝜑
2 nfv 1421 . 2 𝑣𝜑
3 nfv 1421 . 2 𝑥𝜓
4 nfv 1421 . 2 𝑦𝜓
5 cbvoprab12v.1 . 2 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))
61, 2, 3, 4, 5cbvoprab12 5578 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  {coprab 5513 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516 This theorem is referenced by: (None)
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