Theorem exellimddv

Description: Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020)

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

Proof

Step	Hyp	Ref	Expression
1		exellimddv.1	⊢ ( 𝜑 → ∃ 𝑥 𝑥 ∈ 𝐴 )
2		exellimddv.2	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) )
3	2	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
4		exellim	⊢ ( ( ∃ 𝑥 𝑥 ∈ 𝐴 ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) → 𝜓 )
5	1 3 4	syl2anc	⊢ ( 𝜑 → 𝜓 )