Theorem 19.29r 1875

Description: Variation of 19.29 1874. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.)

Assertion

Ref	Expression
19.29r	⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Proof of Theorem 19.29r

Step	Hyp	Ref	Expression
1		pm3.21 470	. . 3 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1832	. 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
3	2	impcom 406	1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1537  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780

This theorem is referenced by:  19.29r2  1876  19.29x  1877  intab  4981  imadif  6631  kmlem6  10152  hashgt23el  14388  2ndcdisj  23180  fmcncfil  33209  bnj907  34276  funen1cnv  34389  loop1cycl  34426  umgr2cycl  34430  bj-19.41al  35839