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Metamath Proof Explorer

Theorem mpjaodan

Description: Eliminate a disjunction in a deduction. A translation of natural deduction rule \/ E ( \/ elimination), see natded . (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	jaodan.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
	jaodan.2	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 )
	jaodan.3	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜃 ) )
Assertion	mpjaodan	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		jaodan.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		jaodan.2	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 )
3		jaodan.3	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜃 ) )
4	1 2	jaodan	⊢ ( ( 𝜑 ∧ ( 𝜓 ∨ 𝜃 ) ) → 𝜒 )
5	3 4	mpdan	⊢ ( 𝜑 → 𝜒 )