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Theorem wl-bitr1 26251
Description: Closed form of bitri 242. Place before bitri 242. [ +33] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
wl-bitr1  |-  ( (
ph 
<->  ps )  ->  (
( ps  <->  ch )  ->  ( ph  <->  ch )
) )

Proof of Theorem wl-bitr1
StepHypRef Expression
1 bi1 180 . . 3  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
2 bi1 180 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
32imim1d 72 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
41, 3syl5 31 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ps  <->  ch )  ->  ( ph  ->  ch ) ) )
5 bi2 191 . . 3  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
6 bi2 191 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
76imim2d 51 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( ch  ->  ps )  ->  ( ch  ->  ph ) ) )
85, 7syl5 31 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ps  <->  ch )  ->  ( ch  ->  ph )
) )
94, 8impbidd 183 1  |-  ( (
ph 
<->  ps )  ->  (
( ps  <->  ch )  ->  ( ph  <->  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178
This theorem is referenced by:  wl-bitri  26252  wl-bitrd  26253  wl-bibi1  26254  wl-bitr  26261  e2ebindALT  29139
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 179
  Copyright terms: Public domain W3C validator