Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.28 Structured version   Visualization version   GIF version

Theorem 19.28 2231
 Description: Theorem 19.28 of [Margaris] p. 90. See 19.28v 1997 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.)
Hypothesis
Ref Expression
19.28.1 𝑥𝜑
Assertion
Ref Expression
19.28 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem 19.28
StepHypRef Expression
1 19.26 1871 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.28.1 . . . 4 𝑥𝜑
3219.3 2203 . . 3 (∀𝑥𝜑𝜑)
43anbi1i 626 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
51, 4bitri 278 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∀wal 1536  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2178 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786 This theorem is referenced by:  aaan  2354  wl-ax11-lem7  34946
 Copyright terms: Public domain W3C validator