Theorem nemtbir 1638

Description: An inference from an inequality, related to modus tollens.

Hypotheses

Ref	Expression
nemtbir.1	|- A =/= B
nemtbir.2	|- (ph <-> A = B)

Assertion

Ref	Expression
nemtbir	|- -. ph

Proof of Theorem nemtbir

Step	Hyp	Ref	Expression
1		nemtbir.1	. . 3 |- A =/= B
2		df-ne 1584	. . 3 |- (A =/= B <-> -. A = B)
3	1, 2	mpbi 189	. 2 |- -. A = B
4		nemtbir.2	. 2 |- (ph <-> A = B)
5	3, 4	mtbir 192	1 |- -. ph

Syntax hints:  -. wn 2   <-> wb 146   = wceq 954   =/= wne 1582

This theorem is referenced by:  opthwiener 2802  snsn0non 3120  opthprc 3216  tz7.44-2 3920  oelim2 4212

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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