Theorem 19.29r 1802

Description: Variation of 19.29 1801. (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		pm3.21 464	. . 3 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1759	. 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
3	2	impcom 446	1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  19.29r2  1803  19.29x  1804  intab  4507  imadif  5973  kmlem6  8977  2ndcdisj  21259  fmcncfil  29977  bnj907  31035  bj-19.41al  32637