Theorem 19.29 1631

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		pm3.2 139	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
2	1	alimi 1466	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → (𝜑 ∧ 𝜓)))
3		exim 1610	. . 3 ⊢ (∀𝑥(𝜓 → (𝜑 ∧ 𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
4	2, 3	syl 14	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
5	4	imp 124	1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.29r  1632  19.29x  1634  19.35-1  1635  equs4  1736  equvini  1769  rexxfrd  4494  funimaexglem  5337  bj-inex  15399