Theorem addrsub

Description: Right-subtraction: Subtraction of the right summand from the result of an addition. (Contributed by BJ, 6-Jun-2019)

	Ref	Expression
Hypotheses	addlsub.a	|- ( ph -> A e. CC )
	addlsub.b	|- ( ph -> B e. CC )
	addlsub.c	|- ( ph -> C e. CC )
Assertion	addrsub	|- ( ph -> ( ( A + B ) = C <-> B = ( C - A ) ) )

Proof

Step	Hyp	Ref	Expression
1		addlsub.a	|- ( ph -> A e. CC )
2		addlsub.b	|- ( ph -> B e. CC )
3		addlsub.c	|- ( ph -> C e. CC )
4	1 2	addcomd	|- ( ph -> ( A + B ) = ( B + A ) )
5	4	eqeq1d	|- ( ph -> ( ( A + B ) = C <-> ( B + A ) = C ) )
6	2 1 3	addlsub	|- ( ph -> ( ( B + A ) = C <-> B = ( C - A ) ) )
7	5 6	bitrd	|- ( ph -> ( ( A + B ) = C <-> B = ( C - A ) ) )