Theorem necon2abiidc 2370

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

Hypothesis

Ref	Expression
necon2abiidc.1	|- (DECID ph -> ( A = B <-> -. ph ) )

Assertion

Ref	Expression
necon2abiidc	|- (DECID ph -> ( ph <-> A =/= B ) )

Proof of Theorem necon2abiidc

Step	Hyp	Ref	Expression
1		necon2abiidc.1	. . . 4 |- (DECID ph -> ( A = B <-> -. ph ) )
2	1	bicomd 140	. . 3 |- (DECID ph -> ( -. ph <-> A = B ) )
3	2	necon1abiidc 2366	. 2 |- (DECID ph -> ( A =/= B <-> ph ) )
4	3	bicomd 140	1 |- (DECID ph -> ( ph <-> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 819   = wceq 1331   =/= wne 2306

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-dc 820  df-ne 2307

This theorem is referenced by: (None)