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Theorem vtocl2ga 2754
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
vtocl2ga.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl2ga.2 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl2ga.3 ((𝑥𝐶𝑦𝐷) → 𝜑)
Assertion
Ref Expression
vtocl2ga ((𝐴𝐶𝐵𝐷) → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜓,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)

Proof of Theorem vtocl2ga
StepHypRef Expression
1 nfcv 2281 . 2 𝑥𝐴
2 nfcv 2281 . 2 𝑦𝐴
3 nfcv 2281 . 2 𝑦𝐵
4 nfv 1508 . 2 𝑥𝜓
5 nfv 1508 . 2 𝑦𝜒
6 vtocl2ga.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
7 vtocl2ga.2 . 2 (𝑦 = 𝐵 → (𝜓𝜒))
8 vtocl2ga.3 . 2 ((𝑥𝐶𝑦𝐷) → 𝜑)
91, 2, 3, 4, 5, 6, 7, 8vtocl2gaf 2753 1 ((𝐴𝐶𝐵𝐷) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  caovcan  5935  genipv  7324  fsumrelem  11247
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