Theorem necon4biddc 2280

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

Hypothesis

Ref	Expression
necon4biddc.1	|- ( ph -> (DECID A = B -> (DECID C = D -> ( A =/= B <-> C =/= D ) ) ) )

Assertion

Ref	Expression
necon4biddc	|- ( ph -> (DECID A = B -> (DECID C = D -> ( A = B <-> C = D ) ) ) )

Proof of Theorem necon4biddc

Step	Hyp	Ref	Expression
1		necon4biddc.1	. . 3 |- ( ph -> (DECID A = B -> (DECID C = D -> ( A =/= B <-> C =/= D ) ) ) )
2		df-ne 2206	. . . 4 |- ( C =/= D <-> -. C = D )
3	2	bibi2i 216	. . 3 |- ( ( A =/= B <-> C =/= D ) <-> ( A =/= B <-> -. C = D ) )
4	1, 3	syl8ib 155	. 2 |- ( ph -> (DECID A = B -> (DECID C = D -> ( A =/= B <-> -. C = D ) ) ) )
5	4	necon4abiddc 2278	1 |- ( ph -> (DECID A = B -> (DECID C = D -> ( A = B <-> C = D ) ) ) )

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-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743  df-ne 2206

This theorem is referenced by:  nebidc  2285