Theorem dedth3h 4524

Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4523. (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 340	. . 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 4523	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜏)
7	2, 6	dedth 4522	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
8	7	3impib 1114	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541  ifcif 4464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-if 4465

This theorem is referenced by:  dedth3v  4527  faclbnd4lem2  13989  dvdsle  16000  gcdaddm  16213  ipdiri  29171  hvaddcan  29411  hvsubadd  29418  norm3dif  29491  omlsii  29744  chjass  29874  ledi  29881  spansncv  29994  pjcjt2  30033  pjopyth  30061  hoaddass  30123  hocsubdir  30126  hoddi  30331