Theorem 19.29r2 1598

Description: Variation of Theorem 19.29 of [Margaris] p. 90 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 1597	. 2 ⊢ ((∃x∃yφ ∧ ∀x∀yψ) → ∃x(∃yφ ∧ ∀yψ))
2		19.29r 1597	. . 3 ⊢ ((∃yφ ∧ ∀yψ) → ∃y(φ ∧ ψ))
3	2	eximi 1576	. 2 ⊢ (∃x(∃yφ ∧ ∀yψ) → ∃x∃y(φ ∧ ψ))
4	1, 3	syl 15	1 ⊢ ((∃x∃yφ ∧ ∀x∀yψ) → ∃x∃y(φ ∧ ψ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃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:  2eu6  2289