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Metamath Proof Explorer

Theorem mpjaod

Description: Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses jaod.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
jaod.2 ⊢ ( 𝜑 → ( 𝜃 → 𝜒 ) )
jaod.3 ⊢ ( 𝜑 → ( 𝜓 ∨ 𝜃 ) )
Assertion mpjaod ( 𝜑 → 𝜒 )

Proof

Step Hyp Ref Expression
1 jaod.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 jaod.2 ⊢ ( 𝜑 → ( 𝜃 → 𝜒 ) )
3 jaod.3 ⊢ ( 𝜑 → ( 𝜓 ∨ 𝜃 ) )
4 1 2 jaod ⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) → 𝜒 ) )
5 3 4 mpd ⊢ ( 𝜑 → 𝜒 )