Theorem rexlimiva

Description: Inference from Theorem 19.23 of Margaris p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006) Shorten dependent theorems. (Revised by Wolf lammen, 23-Dec-2024)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimiva.1	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 )
2		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
3	1	exlimiv	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 )
4	2 3	sylbi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 )