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Theorem dedth4v 4490
Description: Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4488. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
Hypotheses
Ref Expression
dedth4v.1 (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓𝜒))
dedth4v.2 (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒𝜃))
dedth4v.3 (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃𝜏))
dedth4v.4 (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏𝜂))
dedth4v.5 𝜂
Assertion
Ref Expression
dedth4v (𝜑𝜓)

Proof of Theorem dedth4v
StepHypRef Expression
1 dedth4v.1 . . . 4 (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓𝜒))
2 dedth4v.2 . . . 4 (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒𝜃))
3 dedth4v.3 . . . 4 (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃𝜏))
4 dedth4v.4 . . . 4 (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏𝜂))
5 dedth4v.5 . . . 4 𝜂
61, 2, 3, 4, 5dedth4h 4487 . . 3 (((𝜑𝜑) ∧ (𝜑𝜑)) → 𝜓)
76anidms 570 . 2 ((𝜑𝜑) → 𝜓)
87anidms 570 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  ifcif 4428
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 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-if 4429
This theorem is referenced by: (None)
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