Theorem necon1abiddc 2307

Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon1abiddc.1	|- ( ph -> (DECID ps -> ( -. ps <-> A = B ) ) )

Assertion

Ref	Expression
necon1abiddc	|- ( ph -> (DECID ps -> ( A =/= B <-> ps ) ) )

Proof of Theorem necon1abiddc

Step	Hyp	Ref	Expression
1		necon1abiddc.1	. . 3 |- ( ph -> (DECID ps -> ( -. ps <-> A = B ) ) )
2	1	con1biddc 803	. 2 |- ( ph -> (DECID ps -> ( -. A = B <-> ps ) ) )
3		df-ne 2246	. . 3 |- ( A =/= B <-> -. A = B )
4	3	bibi1i 226	. 2 |- ( ( A =/= B <-> ps ) <-> ( -. A = B <-> ps ) )
5	2, 4	syl6ibr 160	1 |- ( ph -> (DECID ps -> ( A =/= B <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103  DECID wdc 775   = wceq 1284   =/= wne 2245

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  df-ne 2246

This theorem is referenced by:  necon2abiddc  2311