Theorem ecase23d 1391

Description: Variation of ecased 1390 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 1010	. . . 4 ⊢ ((𝜓 ∨ 𝜒 ∨ 𝜃) ↔ ((𝜓 ∨ 𝜒) ∨ 𝜃))
5	3, 4	sylib 122	. . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) ∨ 𝜃))
6	2, 5	ecased 1390	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
7	1, 6	ecased 1390	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 720   ∨ w3o 1008

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-3or 1010

This theorem is used by:  iseqf1olemklt  10950  xrmaxiflemcl  12030  xrmaxifle  12031  xrmaxiflemab  12032  xrmaxiflemlub  12033  ennnfonelemex  13357  mulgval  13978  mulgfng  13980  subgmulg  14044