MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.24 Structured version   Visualization version   GIF version

Theorem 19.24 1997
Description: Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
Assertion
Ref Expression
19.24 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Proof of Theorem 19.24
StepHypRef Expression
1 19.2 1986 . . 3 (∀𝑥𝜓 → ∃𝑥𝜓)
21imim2i 16 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
3 19.35 1884 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
42, 3sylibr 237 1 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-6 1975
This theorem depends on definitions:  df-bi 210  df-ex 1787
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator