Theorem dedth4h 4517

Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4515. (Contributed by NM, 16-May-1999.)

Hypotheses

Ref	Expression
dedth4h.1	⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏 ↔ 𝜂))
dedth4h.2	⊢ (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂 ↔ 𝜁))
dedth4h.3	⊢ (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁 ↔ 𝜎))
dedth4h.4	⊢ (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎 ↔ 𝜌))
dedth4h.5	⊢ 𝜌

Assertion

Ref	Expression
dedth4h	⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)

Proof of Theorem dedth4h

Step	Hyp	Ref	Expression
1		dedth4h.1	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏 ↔ 𝜂))
2	1	imbi2d 340	. . 3 ⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (((𝜒 ∧ 𝜃) → 𝜏) ↔ ((𝜒 ∧ 𝜃) → 𝜂)))
3		dedth4h.2	. . . 4 ⊢ (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂 ↔ 𝜁))
4	3	imbi2d 340	. . 3 ⊢ (𝐵 = if(𝜓, 𝐵, 𝑆) → (((𝜒 ∧ 𝜃) → 𝜂) ↔ ((𝜒 ∧ 𝜃) → 𝜁)))
5		dedth4h.3	. . . 4 ⊢ (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁 ↔ 𝜎))
6		dedth4h.4	. . . 4 ⊢ (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎 ↔ 𝜌))
7		dedth4h.5	. . . 4 ⊢ 𝜌
8	5, 6, 7	dedth2h 4515	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜁)
9	2, 4, 8	dedth2h 4515	. 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝜒 ∧ 𝜃) → 𝜏))
10	9	imp 406	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ifcif 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-if 4457

This theorem is referenced by:  dedth4v  4520  fprg  7009  omopth  8452  nn0opth2  13914  ax5seglem8  27207  hvsubsub4  29323  norm3lemt  29415  eigorth  30101