Theorem ecase23d 1309

Description: Variation of ecased 1308 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.)

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	. 2 ⊢ (𝜑 → ¬ 𝜒)
2		ecase23d.2	. . 3 ⊢ (𝜑 → ¬ 𝜃)
3		ecase23d.3	. . . 4 ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))
4		df-3or 944	. . . 4 ⊢ ((𝜓 ∨ 𝜒 ∨ 𝜃) ↔ ((𝜓 ∨ 𝜒) ∨ 𝜃))
5	3, 4	sylib 121	. . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) ∨ 𝜃))
6	2, 5	ecased 1308	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
7	1, 6	ecased 1308	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 680   ∨ w3o 942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 587  ax-io 681

This theorem depends on definitions:  df-bi 116  df-3or 944

This theorem is referenced by:  iseqf1olemklt  10144  xrmaxiflemcl  10899  xrmaxifle  10900  xrmaxiflemab  10901  xrmaxiflemlub  10902  ennnfonelemex  11765