Theorem 19.40-2 1888

Description: Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with two quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
19.40-2	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓))

Proof of Theorem 19.40-2

Step	Hyp	Ref	Expression
1		19.40 1887	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
2	1	eximi 1835	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(∃𝑦𝜑 ∧ ∃𝑦𝜓))
3		19.40 1887	. 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ ∃𝑦𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓))
4	2, 3	syl 17	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓))

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by: (None)