Theorem 19.29 1596

Description: Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Assertion

Ref	Expression
19.29	⊢ ((∀xφ ∧ ∃xψ) → ∃x(φ ∧ ψ))

Proof of Theorem 19.29

Step	Hyp	Ref	Expression
1		pm3.2 434	. . . 4 ⊢ (φ → (ψ → (φ ∧ ψ)))
2	1	alimi 1559	. . 3 ⊢ (∀xφ → ∀x(ψ → (φ ∧ ψ)))
3		exim 1575	. . 3 ⊢ (∀x(ψ → (φ ∧ ψ)) → (∃xψ → ∃x(φ ∧ ψ)))
4	2, 3	syl 15	. 2 ⊢ (∀xφ → (∃xψ → ∃x(φ ∧ ψ)))
5	4	imp 418	1 ⊢ ((∀xφ ∧ ∃xψ) → ∃x(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:  19.29r  1597  19.29x  1599  equs4  1959  equvini  1987  fnfrec  6321