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 2229
Description: Theorem 19.27 of [Margaris] p. 90. See 19.27v 1996 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 1871 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.27.1 . . . 4 𝑥𝜓
3219.3 2202 . . 3 (∀𝑥𝜓𝜓)
43anbi2i 624 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
51, 4bitri 277 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wal 1535  wnf 1784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785
This theorem is referenced by:  aaan  2353
  Copyright terms: Public domain W3C validator