Theorem nonconne 2257

Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Assertion

Ref	Expression
nonconne	|- -. ( A = B /\ A =/= B )

Proof of Theorem nonconne

Step	Hyp	Ref	Expression
1		pm3.24 659	. 2 |- -. ( A = B /\ -. A = B )
2		df-ne 2246	. . 3 |- ( A =/= B <-> -. A = B )
3	2	anbi2i 444	. 2 |- ( ( A = B /\ A =/= B ) <-> ( A = B /\ -. A = B ) )
4	1, 3	mtbir 628	1 |- -. ( A = B /\ A =/= B )

Syntax hints:   -. wn 3   /\ wa 102   = wceq 1284   =/= wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-ne 2246

This theorem is referenced by: (None)