Theorem nncansd

Description: Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 16-Apr-2025)

	Ref	Expression
Hypotheses	nncansd.1	$$⊢\phi \to A\in \mathrm{No}$$
	nncansd.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	nncansd	$$⊢\phi \to A{-}_s\left(A{-}_{s}B\right)=B$$

Proof

Step	Hyp	Ref	Expression
1		nncansd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		nncansd.2	$$⊢\phi \to B\in \mathrm{No}$$
3	1 1 2	subsubs2d	$$⊢\phi \to A{-}_s\left(A{-}_{s}B\right)=A{+}_s\left(B{-}_{s}A\right)$$
4		pncan3s	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to A{+}_s\left(B{-}_{s}A\right)=B$$
5	1 2 4	syl2anc	$$⊢\phi \to A{+}_s\left(B{-}_{s}A\right)=B$$
6	3 5	eqtrd	$$⊢\phi \to A{-}_s\left(A{-}_{s}B\right)=B$$