Theorem rexlimdvv

Description: Inference from Theorem 19.23 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimdvv.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝜓 → 𝜒 ) ) )
2	1	expdimp	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ∈ 𝐵 → ( 𝜓 → 𝜒 ) ) )
3	2	rexlimdv	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ 𝐵 𝜓 → 𝜒 ) )
4	3	rexlimdva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 → 𝜒 ) )