Theorem 19.29 1869

Description: Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1870. (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 1827	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
3	2	imp 406	1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1532  ∃wex 1774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775

This theorem is referenced by:  19.29x  1872  supsrlem  11134  1stccnp  23365  iscmet3  25220  isch3  31050  bnj849  34556  lfuhgr3  34729  axc11n11r  36160  bj-19.42t  36250  stoweidlem35  45423