Theorem 19.29x 1599

Description: Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)

Assertion

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

Proof of Theorem 19.29x

Step	Hyp	Ref	Expression
1		19.29r 1597	. 2 ⊢ ((∃x∀yφ ∧ ∀x∃yψ) → ∃x(∀yφ ∧ ∃yψ))
2		19.29 1596	. . 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

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

This theorem is used by: (None)