Theorem 2r19.29 3139

Description: Theorem r19.29 3114 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 3114	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))
2		r19.29 3114	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓))
3	2	reximi 3084	. 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 395  ∀wral 3061  ∃wrex 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-ral 3062  df-rex 3071

This theorem is referenced by:  r19.29d2r  3140  rnmposs  32684  disjdmqsss  38803  disjdmqscossss  38804  prter2  38882