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Theorem vtocl3ga 3496
 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 2919 . 2 𝑥𝐴
2 nfcv 2919 . 2 𝑦𝐴
3 nfcv 2919 . 2 𝑧𝐴
4 nfcv 2919 . 2 𝑦𝐵
5 nfcv 2919 . 2 𝑧𝐵
6 nfcv 2919 . 2 𝑧𝐶
7 nfv 1915 . 2 𝑥𝜓
8 nfv 1915 . 2 𝑦𝜒
9 nfv 1915 . 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 3495 1 ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-v 3411 This theorem is referenced by:  preq12bg  4741  pocl  5450  jensenlem2  25672  wrdt2ind  30749  xpord3pred  33353
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