MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.9ht Structured version   Visualization version   GIF version

Theorem 19.9ht 2126
Description: A closed version of 19.9 2058. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
Assertion
Ref Expression
19.9ht (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Proof of Theorem 19.9ht
StepHypRef Expression
1 exim 1750 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥𝑥𝜑))
2 axc7e 2116 . 2 (∃𝑥𝑥𝜑𝜑)
31, 2syl6 34 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by:  hbntOLD  2128  19.9dOLD  2189  bj-19.9htbi  31687
  Copyright terms: Public domain W3C validator