Theorem rexlimivv

Description: Inference from Theorem 19.23 of Margaris p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimivv.1	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝜑 → 𝜓 ) )
2	1	rexlimdva	⊢ ( 𝑥 ∈ 𝐴 → ( ∃ 𝑦 ∈ 𝐵 𝜑 → 𝜓 ) )
3	2	rexlimiv	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 → 𝜓 )