Theorem exlimddv

Description: Existential elimination rule of natural deduction (Rule C, explained in exlimiv ). (Contributed by Mario Carneiro, 15-Jun-2016)

	Ref	Expression
Hypotheses	exlimddv.1	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
	exlimddv.2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
Assertion	exlimddv	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		exlimddv.1	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
2		exlimddv.2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
3	2	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4	3	exlimdv	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → 𝜒 ) )
5	1 4	mpd	⊢ ( 𝜑 → 𝜒 )