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Theorem vtocl3ga 2623
 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 2178 . 2 𝑥𝐴
2 nfcv 2178 . 2 𝑦𝐴
3 nfcv 2178 . 2 𝑧𝐴
4 nfcv 2178 . 2 𝑦𝐵
5 nfcv 2178 . 2 𝑧𝐵
6 nfcv 2178 . 2 𝑧𝐶
7 nfv 1421 . 2 𝑥𝜓
8 nfv 1421 . 2 𝑦𝜒
9 nfv 1421 . 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 2622 1 ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393 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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 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 This theorem is referenced by:  preq12bg  3544  pocl  4040  sowlin  4057
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