Theorem con1bdc 805

Description: Contraposition. Bidirectional version of con1dc 786. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
con1bdc	|- (DECID ph -> (DECID ps -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) ) )

Proof of Theorem con1bdc

Step	Hyp	Ref	Expression
1		con1dc 786	. . . 4 |- (DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
2	1	adantr 270	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
3		con1dc 786	. . . 4 |- (DECID ps -> ( ( -. ps -> ph ) -> ( -. ph -> ps ) ) )
4	3	adantl 271	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ps -> ph ) -> ( -. ph -> ps ) ) )
5	2, 4	impbid 127	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) )
6	5	ex 113	1 |- (DECID ph -> (DECID ps -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)