Theorem rexlimdvva

Description: Inference from Theorem 19.23 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimdvva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝜓 → 𝜒 ) )
2	1	ex	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 𝜓 → 𝜒 ) ) )
3	2	rexlimdvv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 → 𝜒 ) )