Theorem 19.29x 1511

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 1509	. 2 ⊢ ((∃x∀yφ ∧ ∀x∃yψ) → ∃x(∀yφ ∧ ∃yψ))
2		19.29 1508	. . 3 ⊢ ((∀yφ ∧ ∃yψ) → ∃y(φ ∧ ψ))
3	2	eximi 1488	. 2 ⊢ (∃x(∀yφ ∧ ∃yψ) → ∃x∃y(φ ∧ ψ))
4	1, 3	syl 14	1 ⊢ ((∃x∀yφ ∧ ∀x∃yψ) → ∃x∃y(φ ∧ ψ))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)