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Theorem vtocl3ga 2751
Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
vtocl3ga.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl3ga.2 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl3ga.3 (𝑧 = 𝐶 → (𝜒𝜃))
vtocl3ga.4 ((𝑥𝐷𝑦𝑅𝑧𝑆) → 𝜑)
Assertion
Ref Expression
vtocl3ga ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝐷,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem vtocl3ga
StepHypRef Expression
1 nfcv 2279 . 2 𝑥𝐴
2 nfcv 2279 . 2 𝑦𝐴
3 nfcv 2279 . 2 𝑧𝐴
4 nfcv 2279 . 2 𝑦𝐵
5 nfcv 2279 . 2 𝑧𝐵
6 nfcv 2279 . 2 𝑧𝐶
7 nfv 1508 . 2 𝑥𝜓
8 nfv 1508 . 2 𝑦𝜒
9 nfv 1508 . 2 𝑧𝜃
10 vtocl3ga.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
11 vtocl3ga.2 . 2 (𝑦 = 𝐵 → (𝜓𝜒))
12 vtocl3ga.3 . 2 (𝑧 = 𝐶 → (𝜒𝜃))
13 vtocl3ga.4 . 2 ((𝑥𝐷𝑦𝑅𝑧𝑆) → 𝜑)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13vtocl3gaf 2750 1 ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 962   = 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 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683
This theorem is referenced by:  preq12bg  3695  pocl  4220  sowlin  4237
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