Theorem rexlimddv

Description: Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimddv.1	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
2		rexlimddv.2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → 𝜒 )
3	2	rexlimdvaa	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )
4	1 3	mpd	⊢ ( 𝜑 → 𝜒 )