Theorem necon1d 2082

Description: Contrapositive law deduction for inequality. (The proof was shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon1d.1	|- (ph -> (A =/= B -> C = D))

Assertion

Ref	Expression
necon1d	|- (ph -> (C =/= D -> A = B))

Proof of Theorem necon1d

Step	Hyp	Ref	Expression
1		necon1d.1	. . 3 |- (ph -> (A =/= B -> C = D))
2		nne 2021	. . 3 |- (-. C =/= D <-> C = D)
3	1, 2	syl6ibr 230	. 2 |- (ph -> (A =/= B -> -. C =/= D))
4	3	necon4ad 2071	1 |- (ph -> (C =/= D -> A = B))

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   =/= wne 2017

This theorem is referenced by:  h1datomi 11137  eigorthi 11400

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-ne 2019

Copyright terms: Public domain