MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.29x Structured version   Visualization version   GIF version

Theorem 19.29x 1801
Description: Variation of 19.29 1798 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
Assertion
Ref Expression
19.29x ((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

Proof of Theorem 19.29x
StepHypRef Expression
1 19.29r 1799 . 2 ((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓))
2 19.29 1798 . . 3 ((∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑦(𝜑𝜓))
32eximi 1759 . 2 (∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
41, 3syl 17 1 ((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator