Theorem 19.29 1900

Description: Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1901. (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		pm3.2 474	. . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1859	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
3	2	imp 411	1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807

This theorem is referenced by:  19.29x  1903  elirrvOLD  9556  supsrlem  11092  1stccnp  23584  iscmet3  25417  isch3  31530  bnj849  35254  lfuhgr3  35507  axc11n11r  37193  bj-19.42t  37275  stoweidlem35  46636