Theorem ecase23d 1469

Description: Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.)

Hypotheses

Ref	Expression
ecase23d.1	⊢ (𝜑 → ¬ 𝜒)
ecase23d.2	⊢ (𝜑 → ¬ 𝜃)
ecase23d.3	⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))

Assertion

Ref	Expression
ecase23d	⊢ (𝜑 → 𝜓)

Proof of Theorem ecase23d

Step	Hyp	Ref	Expression
1		ecase23d.1	. . 3 ⊢ (𝜑 → ¬ 𝜒)
2		ecase23d.2	. . 3 ⊢ (𝜑 → ¬ 𝜃)
3		ioran 981	. . 3 ⊢ (¬ (𝜒 ∨ 𝜃) ↔ (¬ 𝜒 ∧ ¬ 𝜃))
4	1, 2, 3	sylanbrc 581	. 2 ⊢ (𝜑 → ¬ (𝜒 ∨ 𝜃))
5		ecase23d.3	. . . 4 ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))
6		3orass 1087	. . . 4 ⊢ ((𝜓 ∨ 𝜒 ∨ 𝜃) ↔ (𝜓 ∨ (𝜒 ∨ 𝜃)))
7	5, 6	sylib 217	. . 3 ⊢ (𝜑 → (𝜓 ∨ (𝜒 ∨ 𝜃)))
8	7	ord 862	. 2 ⊢ (𝜑 → (¬ 𝜓 → (𝜒 ∨ 𝜃)))
9	4, 8	mt3d 148	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 845   ∨ w3o 1083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085

This theorem is referenced by:  tz7.7  6400  wfrlem10OLD  8345  nolt02o  27648  nogt01o  27649  noresle  27650  nosupbnd1lem6  27666  nosupbnd2lem1  27668  noinfbnd1lem6  27681  sltmul2  28091  archiabllem2b  32925