Theorem necon2abiidc 2269

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 129	. . 3 |- (DECID ph -> ( -. ph <-> A = B ) )
3	2	necon1abiidc 2265	. 2 |- (DECID ph -> ( A =/= B <-> ph ) )
4	3	bicomd 129	1 |- (DECID ph -> ( ph <-> 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: (None)