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

Theorem albi 1820
 Description: Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
Assertion
Ref Expression
albi (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Proof of Theorem albi
StepHypRef Expression
1 biimp 218 . . 3 ((𝜑𝜓) → (𝜑𝜓))
21al2imi 1817 . 2 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3 biimpr 223 . . 3 ((𝜑𝜓) → (𝜓𝜑))
43al2imi 1817 . 2 (∀𝑥(𝜑𝜓) → (∀𝑥𝜓 → ∀𝑥𝜑))
52, 4impbid 215 1 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210 This theorem is referenced by:  albii  1821  nfbiit  1852  albidh  1867  19.16  2225  19.17  2226  equvel  2468  eqeq1d  2760  intmin4  4870  dfiin2g  4924  eunex  5262  bj-2albi  34367  bj-hbxfrbi  34383  bj-sblem  34590  wl-aleq  35246  2albi  41483  ralbidar  41550  trsbcVD  41984  sbcssgVD  41990  ichal  44379
 Copyright terms: Public domain W3C validator