Theorem rexlimiv

Description: Inference from Theorem 19.23 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimiv.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
2	1	rgen	⊢ ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 )
3		r19.23v	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )
4	2 3	mpbi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 )