Theorem necon2ai 1654

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon2ai.1	|- (A = B -> -. ph)

Assertion

Ref	Expression
necon2ai	|- (ph -> A =/= B)

Proof of Theorem necon2ai

Step	Hyp	Ref	Expression
1		necon2ai.1	. . 3 |- (A = B -> -. ph)
2	1	con2i 97	. 2 |- (ph -> -. A = B)
3		df-ne 1630	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	sylibr 198	1 |- (ph -> A =/= B)

Syntax hints:  -. wn 2   -> wi 3   = wceq 992   =/= wne 1628

This theorem is referenced by:  necon2i 1656  intex 2803  iin0 2814  0ellim 3035  pm54.43 4715  inf3lem3 4760  nnne0 6094  vcoprne 8445  strlem1 10458  inficl 11849  totbndbnd 12000

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 145  df-ne 1630

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