MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.16 Structured version   Visualization version   GIF version

Theorem 19.16 2131
Description: Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
Hypothesis
Ref Expression
19.16.1 𝑥𝜑
Assertion
Ref Expression
19.16 (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∀𝑥𝜓))

Proof of Theorem 19.16
StepHypRef Expression
1 19.16.1 . . 3 𝑥𝜑
2119.3 2107 . 2 (∀𝑥𝜑𝜑)
3 albi 1786 . 2 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
42, 3syl5bbr 274 1 (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator