Theorem necon2ai 1611

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 1587	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	sylibr 200	1 |- (ph -> A =/= B)

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   =/= wne 1585

This theorem is referenced by:  necon2i 1613  intex 2729  iin0 2740  0ellim 3031  pm54.43 4572  inf3lem3 4615  nnne0t 5949  vcoprne 8198  strlem1 10177

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 1587

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