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

Theorem ibibr 333
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  |-  ( (
ph  ->  ps )  <->  ( ph  ->  ( ps  <->  ph ) ) )

Proof of Theorem ibibr
StepHypRef Expression
1 pm5.501 331 . . 3  |-  ( ph  ->  ( ps  <->  ( ph  <->  ps ) ) )
2 bicom 192 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ps  <->  ph ) )
31, 2syl6bb 253 . 2  |-  ( ph  ->  ( ps  <->  ( ps  <->  ph ) ) )
43pm5.74i 237 1  |-  ( (
ph  ->  ps )  <->  ( ph  ->  ( ps  <->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177
This theorem is referenced by:  tbt  334  rabxfrd  4744  ufileu  17951  abnotbtaxb  27860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178
  Copyright terms: Public domain W3C validator