Theorem bijadc 872

Description: Combine antecedents into a single biconditional. This inference is reminiscent of jadc 853. (Contributed by Jim Kingdon, 4-May-2018.)

Hypotheses

Ref	Expression
bijadc.1	|- ( ph -> ( ps -> ch ) )
bijadc.2	|- ( -. ph -> ( -. ps -> ch ) )

Assertion

Ref	Expression
bijadc	|- (DECID ps -> ( ( ph <-> ps ) -> ch ) )

Proof of Theorem bijadc

Step	Hyp	Ref	Expression
1		biimpr 129	. . 3 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
2		bijadc.1	. . 3 |- ( ph -> ( ps -> ch ) )
3	1, 2	syli 37	. 2 |- ( ( ph <-> ps ) -> ( ps -> ch ) )
4		biimp 117	. . . 4 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
5	4	con3d 621	. . 3 |- ( ( ph <-> ps ) -> ( -. ps -> -. ph ) )
6		bijadc.2	. . 3 |- ( -. ph -> ( -. ps -> ch ) )
7	5, 6	syli 37	. 2 |- ( ( ph <-> ps ) -> ( -. ps -> ch ) )
8	3, 7	pm2.61ddc 851	1 |- (DECID ps -> ( ( ph <-> ps ) -> ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by: (None)