Theorem dedth3h 4549

Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4548. (Contributed by NM, 15-May-1999.)

Hypotheses

Ref	Expression
dedth3h.1	⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃 ↔ 𝜏))
dedth3h.2	⊢ (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏 ↔ 𝜂))
dedth3h.3	⊢ (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂 ↔ 𝜁))
dedth3h.4	⊢ 𝜁

Assertion

Ref	Expression
dedth3h	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Proof of Theorem dedth3h

Step	Hyp	Ref	Expression
1		dedth3h.1	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃 ↔ 𝜏))
2	1	imbi2d 343	. . 3 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (((𝜓 ∧ 𝜒) → 𝜃) ↔ ((𝜓 ∧ 𝜒) → 𝜏)))
3		dedth3h.2	. . . 4 ⊢ (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏 ↔ 𝜂))
4		dedth3h.3	. . . 4 ⊢ (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂 ↔ 𝜁))
5		dedth3h.4	. . . 4 ⊢ 𝜁
6	3, 4, 5	dedth2h 4548	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜏)
7	2, 6	dedth 4547	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
8	7	3impib 1134	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ifcif 4488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-if 4489

This theorem is referenced by:  dedth3v  4552  faclbnd4lem2  14332  dvdsle  16369  gcdaddm  16584  ipdiri  31163  hvaddcan  31403  hvsubadd  31410  norm3dif  31483  omlsii  31736  chjass  31866  ledi  31873  spansncv  31986  pjcjt2  32025  pjopyth  32053  hoaddass  32115  hocsubdir  32118  hoddi  32323