Theorem 19.29 1871

Description: Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1872. (Contributed by NM, 21-Jun-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		pm3.2 469	. . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1829	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
3	2	imp 406	1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777

This theorem is referenced by:  19.29x  1874  supsrlem  11149  1stccnp  23486  iscmet3  25341  isch3  31270  bnj849  34918  lfuhgr3  35104  axc11n11r  36666  bj-19.42t  36756  stoweidlem35  45991