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Theorem hirstL-ax3 38198
Description: The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.)
Assertion
Ref Expression
hirstL-ax3  |-  ( ( -.  ph  ->  -.  ps )  ->  ( ( -. 
ph  ->  ps )  ->  ph ) )

Proof of Theorem hirstL-ax3
StepHypRef Expression
1 pm4.64 374 . 2  |-  ( ( -.  ph  ->  ps )  <->  (
ph  \/  ps )
)
2 pm4.66 422 . . 3  |-  ( ( -.  ph  ->  -.  ps ) 
<->  ( ph  \/  -.  ps ) )
3 pm2.64 797 . . . 4  |-  ( (
ph  \/  ps )  ->  ( ( ph  \/  -.  ps )  ->  ph )
)
43com12 33 . . 3  |-  ( (
ph  \/  -.  ps )  ->  ( ( ph  \/  ps )  ->  ph )
)
52, 4sylbi 199 . 2  |-  ( ( -.  ph  ->  -.  ps )  ->  ( ( ph  \/  ps )  ->  ph )
)
61, 5syl5bi 221 1  |-  ( ( -.  ph  ->  -.  ps )  ->  ( ( -. 
ph  ->  ps )  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370
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 189  df-or 372
This theorem is referenced by:  ax3h  38199
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