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Theorem biluk 899
Description: Lukasiewicz's shortest axiom for equivalential calculus. Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by NM, 10-Jan-2005.)
Assertion
Ref Expression
biluk  |-  ( (
ph 
<->  ps )  <->  ( ( ch 
<->  ps )  <->  ( ph  <->  ch ) ) )

Proof of Theorem biluk
StepHypRef Expression
1 bicom 191 . . . . 5  |-  ( (
ph 
<->  ps )  <->  ( ps  <->  ph ) )
21bibi1i 305 . . . 4  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ( ps  <->  ph )  <->  ch ) )
3 biass 348 . . . 4  |-  ( ( ( ps  <->  ph )  <->  ch )  <->  ( ps  <->  ( ph  <->  ch )
) )
42, 3bitri 240 . . 3  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ps  <->  ( ph  <->  ch ) ) )
5 biass 348 . . 3  |-  ( ( ( ( ph  <->  ps )  <->  ch )  <->  ( ps  <->  ( ph  <->  ch ) ) )  <->  ( ( ph 
<->  ps )  <->  ( ch  <->  ( ps  <->  ( ph  <->  ch )
) ) ) )
64, 5mpbi 199 . 2  |-  ( (
ph 
<->  ps )  <->  ( ch  <->  ( ps  <->  ( ph  <->  ch )
) ) )
7 biass 348 . 2  |-  ( ( ( ch  <->  ps )  <->  (
ph 
<->  ch ) )  <->  ( ch  <->  ( ps  <->  ( ph  <->  ch )
) ) )
86, 7bitr4i 243 1  |-  ( (
ph 
<->  ps )  <->  ( ( ch 
<->  ps )  <->  ( ph  <->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176
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 177
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