Theorem addsubeq4com

Description: Relation between sums and differences. (Contributed by Steven Nguyen, 5-Jan-2023)

	Ref	Expression
Assertion	addsubeq4com	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( A - C ) = ( D - B ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqcom	|- ( ( A + B ) = ( C + D ) <-> ( C + D ) = ( A + B ) )
2		addsubeq4	|- ( ( ( C e. CC /\ D e. CC ) /\ ( A e. CC /\ B e. CC ) ) -> ( ( C + D ) = ( A + B ) <-> ( A - C ) = ( D - B ) ) )
3	2	ancoms	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) = ( A + B ) <-> ( A - C ) = ( D - B ) ) )
4	1 3	syl5bb	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( A - C ) = ( D - B ) ) )