MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.23h Structured version   Visualization version   GIF version

Theorem 19.23h 2105
Description: Theorem 19.23 of [Margaris] p. 90. See 19.23 2064. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
Hypothesis
Ref Expression
19.23h.1 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
19.23h (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Proof of Theorem 19.23h
StepHypRef Expression
1 19.23h.1 . . 3 (𝜓 → ∀𝑥𝜓)
21nf5i 2009 . 2 𝑥𝜓
3219.23 2064 1 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695  df-nf 1700
This theorem is referenced by:  equsalhw  2106
  Copyright terms: Public domain W3C validator