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Theorem vtocl3ga 3427
 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 2913 . 2 𝑥𝐴
2 nfcv 2913 . 2 𝑦𝐴
3 nfcv 2913 . 2 𝑧𝐴
4 nfcv 2913 . 2 𝑦𝐵
5 nfcv 2913 . 2 𝑧𝐵
6 nfcv 2913 . 2 𝑧𝐶
7 nfv 1995 . 2 𝑥𝜓
8 nfv 1995 . 2 𝑦𝜒
9 nfv 1995 . 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 3426 1 ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353 This theorem is referenced by:  preq12bg  4517  prel12gOLD  4518  pocl  5177  jensenlem2  24935
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