Theorem subsubs2d

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

	Ref	Expression
Hypotheses	subsubs4d.1	$$⊢\phi \to A\in \mathrm{No}$$
	subsubs4d.2	$$⊢\phi \to B\in \mathrm{No}$$
	subsubs4d.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	subsubs2d	$$⊢\phi \to A{-}_s\left(B{-}_{s}C\right)=A{+}_s\left(C{-}_{s}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		subsubs4d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		subsubs4d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		subsubs4d.3	$$⊢\phi \to C\in \mathrm{No}$$
4	2 3	subscld	$$⊢\phi \to B{-}_{s}C\in \mathrm{No}$$
5	1 4	subsvald	$$⊢\phi \to A{-}_s\left(B{-}_{s}C\right)=A{+}_s{+}_s\left(B{-}_{s}C\right)$$
6	2 3	negsubsdi2d	$$⊢\phi \to {+}_s\left(B{-}_{s}C\right)=C{-}_{s}B$$
7	6	oveq2d	$$⊢\phi \to A{+}_s{+}_s\left(B{-}_{s}C\right)=A{+}_s\left(C{-}_{s}B\right)$$
8	5 7	eqtrd	$$⊢\phi \to A{-}_s\left(B{-}_{s}C\right)=A{+}_s\left(C{-}_{s}B\right)$$