Theorem efsubd

Description: Difference of exponents law for exponential function, deduction form. (Contributed by SN, 25-Apr-2025)

	Ref	Expression
Hypotheses	efsubd.a	|- ( ph -> A e. CC )
	efsubd.b	|- ( ph -> B e. CC )
Assertion	efsubd	|- ( ph -> ( exp ` ( A - B ) ) = ( ( exp ` A ) / ( exp ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		efsubd.a	|- ( ph -> A e. CC )
2		efsubd.b	|- ( ph -> B e. CC )
3		efsub	|- ( ( A e. CC /\ B e. CC ) -> ( exp ` ( A - B ) ) = ( ( exp ` A ) / ( exp ` B ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( exp ` ( A - B ) ) = ( ( exp ` A ) / ( exp ` B ) ) )