Theorem con1bdc 772

Description: Contraposition. Bidirectional version of con1dc 753. (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 753	. . . 4 |- (DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
2	1	adantr 261	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
3		con1dc 753	. . . 4 |- (DECID ps -> ( ( -. ps -> ph ) -> ( -. ph -> ps ) ) )
4	3	adantl 262	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ps -> ph ) -> ( -. ph -> ps ) ) )
5	2, 4	impbid 120	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) )
6	5	ex 108	1 |- (DECID ph -> (DECID ps -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)