MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.27 Structured version   Visualization version   GIF version

Theorem 19.27 2270
Description: Theorem 19.27 of [Margaris] p. 90. See 19.27v 2094 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
19.27.1 𝑥𝜓
Assertion
Ref Expression
19.27 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Proof of Theorem 19.27
StepHypRef Expression
1 19.26 1972 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.27.1 . . . 4 𝑥𝜓
3219.3 2243 . . 3 (∀𝑥𝜓𝜓)
43anbi2i 616 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
51, 4bitri 267 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  wal 1654  wnf 1882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-nf 1883
This theorem is referenced by:  aaan  2363
  Copyright terms: Public domain W3C validator