Theorem con1bdc 863

Description: Contraposition. Bidirectional version of con1dc 841. (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 841	. . . 4 |- (DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
2	1	adantr 274	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
3		con1dc 841	. . . 4 |- (DECID ps -> ( ( -. ps -> ph ) -> ( -. ph -> ps ) ) )
4	3	adantl 275	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ps -> ph ) -> ( -. ph -> ps ) ) )
5	2, 4	impbid 128	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) )
6	5	ex 114	1 |- (DECID ph -> (DECID ps -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by: (None)