Theorem hirstL-ax3 41059

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

Step	Hyp	Ref	Expression
1		pm4.64 387	. 2 |- ( ( -. ph -> ps ) <-> ( ph \/ ps ) )
2		pm4.66 436	. . 3 |- ( ( -. ph -> -. ps ) <-> ( ph \/ -. ps ) )
3		pm2.64 830	. . . 4 |- ( ( ph \/ ps ) -> ( ( ph \/ -. ps ) -> ph ) )
4	3	com12 32	. . 3 |- ( ( ph \/ -. ps ) -> ( ( ph \/ ps ) -> ph ) )
5	2, 4	sylbi 207	. 2 |- ( ( -. ph -> -. ps ) -> ( ( ph \/ ps ) -> ph ) )
6	1, 5	syl5bi 232	1 |- ( ( -. ph -> -. ps ) -> ( ( -. ph -> ps ) -> ph ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383

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 197  df-or 385

This theorem is referenced by:  ax3h  41060