Theorem nebidc 2447

Description: Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)

Assertion

Ref	Expression
nebidc	|- (DECID A = B -> (DECID C = D -> ( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) ) ) )

Proof of Theorem nebidc

Step	Hyp	Ref	Expression
1		id 19	. . . 4 |- ( ( A = B <-> C = D ) -> ( A = B <-> C = D ) )
2	1	necon3bid 2408	. . 3 |- ( ( A = B <-> C = D ) -> ( A =/= B <-> C =/= D ) )
3		id 19	. . . . . . . 8 |- ( ( A =/= B <-> C =/= D ) -> ( A =/= B <-> C =/= D ) )
4	3	a1d 22	. . . . . . 7 |- ( ( A =/= B <-> C =/= D ) -> (DECID C = D -> ( A =/= B <-> C =/= D ) ) )
5	4	a1d 22	. . . . . 6 |- ( ( A =/= B <-> C =/= D ) -> (DECID A = B -> (DECID C = D -> ( A =/= B <-> C =/= D ) ) ) )
6	5	necon4biddc 2442	. . . . 5 |- ( ( A =/= B <-> C =/= D ) -> (DECID A = B -> (DECID C = D -> ( A = B <-> C = D ) ) ) )
7	6	com3l 81	. . . 4 |- (DECID A = B -> (DECID C = D -> ( ( A =/= B <-> C =/= D ) -> ( A = B <-> C = D ) ) ) )
8	7	imp 124	. . 3 |- ( (DECID A = B /\ DECID C = D ) -> ( ( A =/= B <-> C =/= D ) -> ( A = B <-> C = D ) ) )
9	2, 8	impbid2 143	. 2 |- ( (DECID A = B /\ DECID C = D ) -> ( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) ) )
10	9	ex 115	1 |- (DECID A = B -> (DECID C = D -> ( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   = wceq 1364   =/= wne 2367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-ne 2368

This theorem is referenced by:  rpexp  12331