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

Theorem ibibr 369
Description: Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
Assertion
Ref Expression
ibibr ((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))

Proof of Theorem ibibr
StepHypRef Expression
1 pm5.501 367 . . 3 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2 bicom 221 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
31, 2bitrdi 287 . 2 (𝜑 → (𝜓 ↔ (𝜓𝜑)))
43pm5.74i 270 1 ((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 206
This theorem is referenced by:  tbt  370  rabxfrd  5344  ufileu  23068  abnotbtaxb  44378
  Copyright terms: Public domain W3C validator