Theorem pncan3s

Description: Subtraction and addition of equals. (Contributed by Scott Fenton, 4-Feb-2025)

	Ref	Expression
Assertion	pncan3s	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to A{+}_s\left(B{-}_{s}A\right)=B$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢B{-}_{s}A=B{-}_{s}A$$
2		simpr	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to B\in \mathrm{No}$$
3		simpl	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to A\in \mathrm{No}$$
4	2 3	subscld	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to B{-}_{s}A\in \mathrm{No}$$
5	2 3 4	subaddsd	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(B{-}_{s}A=B{-}_{s}A\leftrightarrow A{+}_s\left(B{-}_{s}A\right)=B\right)$$
6	1 5	mpbii	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to A{+}_s\left(B{-}_{s}A\right)=B$$