Theorem rexlimdva2

Description: Inference from Theorem 19.23 of Margaris p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypothesis	rexlimdva2.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
Assertion	rexlimdva2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rexlimdva2.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
2	1	exp31	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
3	2	rexlimdv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )