Theorem ecase33d 1501

Description: Deduction for elimination by cases. (Contributed by Thierry Arnoux, 5-Jul-2026.)

Hypotheses

Ref	Expression
ecase33d.1	⊢ (𝜑 → ¬ 𝜓)
ecase33d.2	⊢ (𝜑 → ¬ 𝜒)
ecase33d.3	⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))

Assertion

Ref	Expression
ecase33d	⊢ (𝜑 → 𝜃)

Proof of Theorem ecase33d

Step	Hyp	Ref	Expression
1		ecase33d.3	. . 3 ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))
2		df-3or 1102	. . 3 ⊢ ((𝜓 ∨ 𝜒 ∨ 𝜃) ↔ ((𝜓 ∨ 𝜒) ∨ 𝜃))
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) ∨ 𝜃))
4		ecase33d.1	. . 3 ⊢ (𝜑 → ¬ 𝜓)
5		ecase33d.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
6		ioran 999	. . 3 ⊢ (¬ (𝜓 ∨ 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))
7	4, 5, 6	sylanbrc 594	. 2 ⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒))
8	3, 7	orcnd 891	1 ⊢ (𝜑 → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 860   ∨ w3o 1100

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 210  df-an 401  df-or 861  df-3or 1102

This theorem is referenced by:  nhpmirhp  29034