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Theorem ax2 1422
Description: Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax2  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )

Proof of Theorem ax2
StepHypRef Expression
1 luklem7 1419 . 2  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
2 luklem8 1420 . . 3  |-  ( ( ps  ->  ( ph  ->  ch ) )  -> 
( ( ph  ->  ps )  ->  ( ph  ->  ( ph  ->  ch ) ) ) )
3 luklem6 1418 . . . 4  |-  ( (
ph  ->  ( ph  ->  ch ) )  ->  ( ph  ->  ch ) )
4 luklem8 1420 . . . 4  |-  ( ( ( ph  ->  ( ph  ->  ch ) )  ->  ( ph  ->  ch ) )  ->  (
( ( ph  ->  ps )  ->  ( ph  ->  ( ph  ->  ch ) ) )  -> 
( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) ) )
53, 4ax-mp 8 . . 3  |-  ( ( ( ph  ->  ps )  ->  ( ph  ->  (
ph  ->  ch ) ) )  ->  ( ( ph  ->  ps )  -> 
( ph  ->  ch )
) )
62, 5luklem1 1413 . 2  |-  ( ( ps  ->  ( ph  ->  ch ) )  -> 
( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
71, 6luklem1 1413 1  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
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