Theorem necon2ai 1614

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 1590	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	sylibr 200	1 |- (ph -> A =/= B)

Syntax hints:  -. wn 2   -> wi 3   = wceq 958   =/= wne 1588

This theorem is referenced by:  necon2i 1616  intex 2734  iin0 2745  0ellim 3037  pm54.43 4581  inf3lem3 4624  nnne0t 5951  vcoprne 8194  strlem1 10172

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 1590

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