Theorem repncan2

Description: Addition and subtraction of equals. Compare pncan2 . (Contributed by Steven Nguyen, 8-Jan-2023)

	Ref	Expression
Assertion	repncan2	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A+B\right){-}_{ℝ}A=B$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢A+B=A+B$$
2		readdcl	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A+B\in ℝ$$
3		simpl	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A\in ℝ$$
4		simpr	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to B\in ℝ$$
5	2 3 4	resubaddd	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(\left(A+B\right){-}_{ℝ}A=B\leftrightarrow A+B=A+B\right)$$
6	1 5	mpbiri	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A+B\right){-}_{ℝ}A=B$$