Theorem dedth4v 4489

Description: Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4487. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Hypotheses

Ref	Expression
dedth4v.1	⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓 ↔ 𝜒))
dedth4v.2	⊢ (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒 ↔ 𝜃))
dedth4v.3	⊢ (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃 ↔ 𝜏))
dedth4v.4	⊢ (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏 ↔ 𝜂))
dedth4v.5	⊢ 𝜂

Assertion

Ref	Expression
dedth4v	⊢ (𝜑 → 𝜓)

Proof of Theorem dedth4v

Step	Hyp	Ref	Expression
1		dedth4v.1	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓 ↔ 𝜒))
2		dedth4v.2	. . . 4 ⊢ (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒 ↔ 𝜃))
3		dedth4v.3	. . . 4 ⊢ (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃 ↔ 𝜏))
4		dedth4v.4	. . . 4 ⊢ (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏 ↔ 𝜂))
5		dedth4v.5	. . . 4 ⊢ 𝜂
6	1, 2, 3, 4, 5	dedth4h 4486	. . 3 ⊢ (((𝜑 ∧ 𝜑) ∧ (𝜑 ∧ 𝜑)) → 𝜓)
7	6	anidms 570	. 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓)
8	7	anidms 570	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  ifcif 4425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-if 4426

This theorem is referenced by: (None)