Theorem necon4i 2829

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon4i.1	|- ( A =/= B -> C =/= D )

Assertion

Ref	Expression
necon4i	|- ( C = D -> A = B )

Proof of Theorem necon4i

Step	Hyp	Ref	Expression
1		necon4i.1	. . 3 |- ( A =/= B -> C =/= D )
2	1	neneqd 2799	. 2 |- ( A =/= B -> -. C = D )
3	2	necon4ai 2825	1 |- ( C = D -> A = B )

Syntax hints:   -> wi 4   = 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:  unixp0  5669  scott0  8749  nn0opthi  13057