Theorem necon1abiidc 2401

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

Hypothesis

Ref	Expression
necon1abiidc.1	|- (DECID ph -> ( -. ph <-> A = B ) )

Assertion

Ref	Expression
necon1abiidc	|- (DECID ph -> ( A =/= B <-> ph ) )

Proof of Theorem necon1abiidc

Step	Hyp	Ref	Expression
1		df-ne 2342	. 2 |- ( A =/= B <-> -. A = B )
2		necon1abiidc.1	. . 3 |- (DECID ph -> ( -. ph <-> A = B ) )
3	2	con1biidc 873	. 2 |- (DECID ph -> ( -. A = B <-> ph ) )
4	1, 3	bitrid 191	1 |- (DECID ph -> ( A =/= B <-> ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 830   = wceq 1349   =/= wne 2341

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 610  ax-in2 611  ax-io 705

This theorem depends on definitions:  df-bi 116  df-dc 831  df-ne 2342

This theorem is referenced by:  necon2abiidc  2405