Theorem 19.29r 1871

Description: Variation of 19.29 1870. (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		pm3.21 474	. . 3 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1828	. 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
3	2	impcom 410	1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by:  19.29r2  1872  19.29x  1873  intab  4898  imadif  6432  kmlem6  9575  hashgt23el  13779  2ndcdisj  22058  fmcncfil  31169  bnj907  32234  funen1cnv  32352  loop1cycl  32379  umgr2cycl  32383  bj-19.41al  33987