Theorem necon3i 1602

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 59	. . 3 |- ((A = B -> C = D) -> (A = B -> C = D))
3	2	necon3d 1601	. 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 954   =/= wne 1582

This theorem is referenced by:  onnev 3237  xpnz 3458  unixp 3509  inf3lem2 4594  infeq5 4601  ivthlem8 7231  ivthlem8OLD 7240  nmlno0lem 8398  blocni 8409  nmlnop0ALT 9858  fiiu2 10413

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 147  df-ne 1584

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