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Theorem vtocl4g 3267
Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)
Hypotheses
Ref Expression
vtocl4ga.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl4ga.2 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl4ga.3 (𝑧 = 𝐶 → (𝜒𝜌))
vtocl4ga.4 (𝑤 = 𝐷 → (𝜌𝜃))
vtocl4g.5 𝜑
Assertion
Ref Expression
vtocl4g (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇)) → 𝜃)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑦,𝑧   𝑤,𝐶,𝑧   𝑤,𝐷   𝑤,𝑅,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧   𝑤,𝑇,𝑥,𝑦,𝑧   𝑤,𝑄,𝑥,𝑦,𝑧   𝜓,𝑥   𝜌,𝑧   𝜃,𝑤   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑦,𝑧,𝑤)   𝜒(𝑥,𝑧,𝑤)   𝜃(𝑥,𝑦,𝑧)   𝜌(𝑥,𝑦,𝑤)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦,𝑧)

Proof of Theorem vtocl4g
StepHypRef Expression
1 vtocl4ga.3 . . . 4 (𝑧 = 𝐶 → (𝜒𝜌))
21imbi2d 330 . . 3 (𝑧 = 𝐶 → (((𝐴𝑄𝐵𝑅) → 𝜒) ↔ ((𝐴𝑄𝐵𝑅) → 𝜌)))
3 vtocl4ga.4 . . . 4 (𝑤 = 𝐷 → (𝜌𝜃))
43imbi2d 330 . . 3 (𝑤 = 𝐷 → (((𝐴𝑄𝐵𝑅) → 𝜌) ↔ ((𝐴𝑄𝐵𝑅) → 𝜃)))
5 vtocl4ga.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
6 vtocl4ga.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
7 vtocl4g.5 . . . 4 𝜑
85, 6, 7vtocl2g 3260 . . 3 ((𝐴𝑄𝐵𝑅) → 𝜒)
92, 4, 8vtocl2g 3260 . 2 ((𝐶𝑆𝐷𝑇) → ((𝐴𝑄𝐵𝑅) → 𝜃))
109impcom 446 1 (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇)) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192
This theorem is referenced by:  vtocl4ga  3268
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