Theorem 19.40-2 1610

Description: Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 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 1609	. . 3 ⊢ (∃y(φ ∧ ψ) → (∃yφ ∧ ∃yψ))
2	1	eximi 1576	. 2 ⊢ (∃x∃y(φ ∧ ψ) → ∃x(∃yφ ∧ ∃yψ))
3		19.40 1609	. 2 ⊢ (∃x(∃yφ ∧ ∃yψ) → (∃x∃yφ ∧ ∃x∃yψ))
4	2, 3	syl 15	1 ⊢ (∃x∃y(φ ∧ ψ) → (∃x∃yφ ∧ ∃x∃yψ))

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)