Theorem dedth3h 4589

Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4588. (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 341	. . 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 4588	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜏)
7	2, 6	dedth 4587	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
8	7	3impib 1117	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ifcif 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-if 4530

This theorem is referenced by:  dedth3v  4592  faclbnd4lem2  14254  dvdsle  16253  gcdaddm  16466  ipdiri  30114  hvaddcan  30354  hvsubadd  30361  norm3dif  30434  omlsii  30687  chjass  30817  ledi  30824  spansncv  30937  pjcjt2  30976  pjopyth  31004  hoaddass  31066  hocsubdir  31069  hoddi  31274