Theorem divne1d

Description: If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	div1d.1	|- ( ph -> A e. CC )
	divcld.2	|- ( ph -> B e. CC )
	divcld.3	|- ( ph -> B =/= 0 )
	divne1d.4	|- ( ph -> A =/= B )
Assertion	divne1d	|- ( ph -> ( A / B ) =/= 1 )

Proof

Step	Hyp	Ref	Expression
1		div1d.1	|- ( ph -> A e. CC )
2		divcld.2	|- ( ph -> B e. CC )
3		divcld.3	|- ( ph -> B =/= 0 )
4		divne1d.4	|- ( ph -> A =/= B )
5	1 2 3	diveq1ad	|- ( ph -> ( ( A / B ) = 1 <-> A = B ) )
6	5	necon3bid	|- ( ph -> ( ( A / B ) =/= 1 <-> A =/= B ) )
7	4 6	mpbird	|- ( ph -> ( A / B ) =/= 1 )