Theorem 2r19.29

Description: Theorem r19.29 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010)

	Ref	Expression
Assertion	2r19.29	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.29	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) )
2		r19.29	⊢ ( ( ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) )
3	2	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) )
4	1 3	syl	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) )