Theorem 19.29 1874

Description: Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1875. (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		pm3.2 473	. . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1833	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
3	2	imp 410	1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782

This theorem is referenced by:  19.29x  1877  supsrlem  10522  1stccnp  22067  iscmet3  23897  isch3  29024  bnj849  32307  lfuhgr3  32479  axc11n11r  34130  bj-19.42t  34217  stoweidlem35  42677