Theorem dedth3h 4546

Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4545. (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 4545	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜏)
7	2, 6	dedth 4544	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
8	7	3impib 1134	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ifcif 4485

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 2147  ax-9 2155  ax-ext 2734

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 2741  df-cleq 2754  df-clel 2837  df-if 4486

This theorem is used by:  dedth3v  4549  faclbnd4lem2  14362  dvdsle  16406  gcdaddm  16621  ipdiri  31319  hvaddcan  31559  hvsubadd  31566  norm3dif  31639  omlsii  31892  chjass  32022  ledi  32029  spansncv  32142  pjcjt2  32181  pjopyth  32209  hoaddass  32271  hocsubdir  32274  hoddi  32479