Theorem necon1bbiddc 2268

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

Hypothesis

Ref	Expression
necon1bbiddc.1	|- ( ph -> (DECID A = B -> ( A =/= B <-> ps ) ) )

Assertion

Ref	Expression
necon1bbiddc	|- ( ph -> (DECID A = B -> ( -. ps <-> A = B ) ) )

Proof of Theorem necon1bbiddc

Step	Hyp	Ref	Expression
1		necon1bbiddc.1	. . 3 |- ( ph -> (DECID A = B -> ( A =/= B <-> ps ) ) )
2		df-ne 2206	. . . 4 |- ( A =/= B <-> -. A = B )
3	2	bibi1i 217	. . 3 |- ( ( A =/= B <-> ps ) <-> ( -. A = B <-> ps ) )
4	1, 3	syl6ib 150	. 2 |- ( ph -> (DECID A = B -> ( -. A = B <-> ps ) ) )
5	4	con1biddc 770	1 |- ( ph -> (DECID A = B -> ( -. ps <-> A = B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98  DECID wdc 742   = wceq 1243   =/= wne 2204

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  df-ne 2206

This theorem is referenced by:  necon2bbiddc  2272