Theorem ecase23d 1503

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.3	. . 3 ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))
2		3orass 1106	. . 3 ⊢ ((𝜓 ∨ 𝜒 ∨ 𝜃) ↔ (𝜓 ∨ (𝜒 ∨ 𝜃)))
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → (𝜓 ∨ (𝜒 ∨ 𝜃)))
4		ecase23d.1	. . 3 ⊢ (𝜑 → ¬ 𝜒)
5		ecase23d.2	. . 3 ⊢ (𝜑 → ¬ 𝜃)
6		ioran 999	. . 3 ⊢ (¬ (𝜒 ∨ 𝜃) ↔ (¬ 𝜒 ∧ ¬ 𝜃))
7	4, 5, 6	sylanbrc 595	. 2 ⊢ (𝜑 → ¬ (𝜒 ∨ 𝜃))
8	3, 7	olcnd 891	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 861   ∨ w3o 1102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104

This theorem is used by:  tz7.7  6390  chnpof1  18708  nolt02o  27910  nogt01o  27911  noresle  27912  nosupbnd1lem6  27928  nosupbnd2lem1  27930  noinfbnd1lem6  27943  ltmuls2  28415  rexmul2  33169  archiabllem2b  33580