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

Step	Hyp	Ref	Expression
1		bi1 180	. . 3 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
2		bi1 180	. . . 4 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
3	2	imim1d 72	. . 3 |- ( ( ph <-> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )
4	1, 3	syl5 31	. 2 |- ( ( ph <-> ps ) -> ( ( ps <-> ch ) -> ( ph -> ch ) ) )
5		bi2 191	. . 3 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
6		bi2 191	. . . 4 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
7	6	imim2d 51	. . 3 |- ( ( ph <-> ps ) -> ( ( ch -> ps ) -> ( ch -> ph ) ) )
8	5, 7	syl5 31	. 2 |- ( ( ph <-> ps ) -> ( ( ps <-> ch ) -> ( ch -> ph ) ) )
9	4, 8	impbidd 183	1 |- ( ( ph <-> ps ) -> ( ( ps <-> ch ) -> ( ph <-> ch ) ) )

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