Theorem dedth3v 4553

Description: Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4552. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Hypotheses

Ref	Expression
dedth3v.1	⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓 ↔ 𝜒))
dedth3v.2	⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃))
dedth3v.3	⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏))
dedth3v.4	⊢ 𝜏

Assertion

Ref	Expression
dedth3v	⊢ (𝜑 → 𝜓)

Proof of Theorem dedth3v

Step	Hyp	Ref	Expression
1		dedth3v.1	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓 ↔ 𝜒))
2		dedth3v.2	. . . 4 ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃))
3		dedth3v.3	. . . 4 ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏))
4		dedth3v.4	. . . 4 ⊢ 𝜏
5	1, 2, 3, 4	dedth3h 4550	. . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) → 𝜓)
6	5	3anidm12 1446	. 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓)
7	6	anidms 577	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ifcif 4489

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-if 4490

This theorem is used by:  sseliALT  5274