Theorem 2r19.29 3289

Description: Theorem r19.29 3282 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Assertion

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

Proof of Theorem 2r19.29

Step	Hyp	Ref	Expression
1		r19.29 3282	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))
2		r19.29 3282	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓))
3	2	reximi 3219	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓))
4	1, 3	syl 17	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 386  ∀wral 3117  ∃wrex 3118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-ral 3122  df-rex 3123

This theorem is referenced by:  prter2  34956