Theorem 19.29r2 1870

Description: Variation of 19.29r 1869 with double quantification. (Contributed by NM, 3-Feb-2005.)

Assertion

Ref	Expression
19.29r2	⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))

Proof of Theorem 19.29r2

Step	Hyp	Ref	Expression
1		19.29r 1869	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥(∃𝑦𝜑 ∧ ∀𝑦𝜓))
2		19.29r 1869	. . 3 ⊢ ((∃𝑦𝜑 ∧ ∀𝑦𝜓) → ∃𝑦(𝜑 ∧ 𝜓))
3	2	eximi 1829	. 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ ∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))
4	1, 3	syl 17	1 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1531  ∃wex 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774

This theorem is referenced by:  funen1cnv  34744