Theorem resubaddd

Description: Relationship between subtraction and addition. Based on subaddd . (Contributed by Steven Nguyen, 8-Jan-2023)

	Ref	Expression
Hypotheses	resubaddd.1	|- ( ph -> A e. RR )
	resubaddd.2	|- ( ph -> B e. RR )
	resubaddd.3	|- ( ph -> C e. RR )
Assertion	resubaddd	|- ( ph -> ( ( A -R B ) = C <-> ( B + C ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		resubaddd.1	|- ( ph -> A e. RR )
2		resubaddd.2	|- ( ph -> B e. RR )
3		resubaddd.3	|- ( ph -> C e. RR )
4		resubadd	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A -R B ) = C <-> ( B + C ) = A ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( ( A -R B ) = C <-> ( B + C ) = A ) )