Theorem dedth3v 4144

Description: Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4143. (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 4141	. . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) → 𝜓)
6	5	3anidm12 1383	. 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓)
7	6	anidms 677	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  sseliALT  4791