Theorem necon3i 2042

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon3i.1	|- (A = B -> C = D)

Assertion

Ref	Expression
necon3i	|- (C =/= D -> A =/= B)

Proof of Theorem necon3i

Step	Hyp	Ref	Expression
1		necon3i.1	. 2 |- (A = B -> C = D)
2		id 73	. . 3 |- ((A = B -> C = D) -> (A = B -> C = D))
3	2	necon3d 2041	. 2 |- ((A = B -> C = D) -> (C =/= D -> A =/= B))
4	1, 3	ax-mp 7	1 |- (C =/= D -> A =/= B)

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

This theorem is referenced by:  onnev 4070  xpnz 4335  unixp 4422  inf3lem2 5720  infeq5 5727  ivthlem8 8550  nmlno0lem 9793  blocni 9805  fiiu2 10220  nmlnop0iALT 11557  nepss 13820

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