Theorem nonconne 2352

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		pm3.24 688	. 2 |- -. ( A = B /\ -. A = B )
2		df-ne 2341	. . 3 |- ( A =/= B <-> -. A = B )
3	2	anbi2i 454	. 2 |- ( ( A = B /\ A =/= B ) <-> ( A = B /\ -. A = B ) )
4	1, 3	mtbir 666	1 |- -. ( A = B /\ A =/= B )

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1348   =/= wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-ne 2341

This theorem is referenced by: (None)