Theorem rexlimdv

Description: Inference from Theorem 19.23 of Margaris p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002) (Proof shortened by Eric Schmidt, 22-Dec-2006) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020)

	Ref	Expression
Hypothesis	rexlimdv.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
Assertion	rexlimdv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rexlimdv.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
2	1	com3l	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
3	2	rexlimiv	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 → ( 𝜑 → 𝜒 ) )
4	3	com12	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )