Theorem 3o2cs 32409

Description: Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Hypothesis

Ref	Expression
3o1cs.1	⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃)

Assertion

Ref	Expression
3o2cs	⊢ (𝜓 → 𝜃)

Proof of Theorem 3o2cs

Step	Hyp	Ref	Expression
1		df-3or 1087	. . . 4 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
2		3o1cs.1	. . . 4 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃)
3	1, 2	sylbir 235	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → 𝜃)
4	3	orcs 875	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜃)
5	4	olcs 876	1 ⊢ (𝜓 → 𝜃)

Syntax hints:   → wi 4   ∨ wo 847   ∨ w3o 1085

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 207  df-or 848  df-3or 1087

This theorem is referenced by:  xrpxdivcld  32857