Theorem 19.29 1608

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		pm3.2 138	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
2	1	alimi 1443	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → (𝜑 ∧ 𝜓)))
3		exim 1587	. . 3 ⊢ (∀𝑥(𝜓 → (𝜑 ∧ 𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
4	2, 3	syl 14	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
5	4	imp 123	1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1341  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.29r  1609  19.29x  1611  19.35-1  1612  equs4  1713  equvini  1746  rexxfrd  4441  funimaexglem  5271  bj-inex  13789