Theorem subrecd

Description: Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017)

	Ref	Expression
Hypotheses	subrecd.1	$$⊢\phi \to A\in ℂ$$
	subrecd.2	$$⊢\phi \to B\in ℂ$$
	subrecd.3	$$⊢\phi \to A\ne 0$$
	subrecd.4	$$⊢\phi \to B\ne 0$$
Assertion	subrecd	$$⊢\phi \to \left(\frac{1}{A}\right)-\left(\frac{1}{B}\right)=\frac{B-A}{AB}$$

Proof

Step	Hyp	Ref	Expression
1		subrecd.1	$$⊢\phi \to A\in ℂ$$
2		subrecd.2	$$⊢\phi \to B\in ℂ$$
3		subrecd.3	$$⊢\phi \to A\ne 0$$
4		subrecd.4	$$⊢\phi \to B\ne 0$$
5		subrec	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{1}{A}\right)-\left(\frac{1}{B}\right)=\frac{B-A}{AB}$$
6	1 3 2 4 5	syl22anc	$$⊢\phi \to \left(\frac{1}{A}\right)-\left(\frac{1}{B}\right)=\frac{B-A}{AB}$$