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Metamath Proof Explorer

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	imp	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 )
3	2	rexlimiva	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 )