Theorem necon1bbii 2843

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon1bbii.1	|- ( A =/= B <-> ph )

Assertion

Ref	Expression
necon1bbii	|- ( -. ph <-> A = B )

Proof of Theorem necon1bbii

Step	Hyp	Ref	Expression
1		nne 2798	. 2 |- ( -. A =/= B <-> A = B )
2		necon1bbii.1	. 2 |- ( A =/= B <-> ph )
3	1, 2	xchnxbi 322	1 |- ( -. ph <-> A = B )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-ne 2795

This theorem is referenced by:  necon2bbii  2845  rabeq0OLD  3960  intnex  4821  class2set  4832  csbopab  5008  relimasn  5488  modom  8161  supval2  8361  fzo0  12492  vma1  24892  lgsquadlem3  25107  ordtconnlem1  29970