Theorem rexlimi

Description: Restricted quantifier version of exlimi . For a version based on fewer axioms see rexlimiv . (Contributed by NM, 30-Nov-2003) (Proof shortened by Andrew Salmon, 30-May-2011)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimi.1	⊢ Ⅎ 𝑥 𝜓
2		rexlimi.2	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
3	2	rgen	⊢ ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 )
4	1	r19.23	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )
5	3 4	mpbi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 )