Theorem addsubs4d

Description: Rearrangement of four terms in mixed addition and subtraction. Surreal version. (Contributed by Scott Fenton, 25-Jul-2025)

	Ref	Expression
Hypotheses	addsubs4d.1	$$⊢\phi \to A\in \mathrm{No}$$
	addsubs4d.2	$$⊢\phi \to B\in \mathrm{No}$$
	addsubs4d.3	$$⊢\phi \to C\in \mathrm{No}$$
	addsubs4d.4	$$⊢\phi \to D\in \mathrm{No}$$
Assertion	addsubs4d	$$⊢\phi \to \left(A{+}_{s}B\right){-}_s\left(C{+}_{s}D\right)=\left(A{-}_{s}C\right){+}_s\left(B{-}_{s}D\right)$$

Proof

Step	Hyp	Ref	Expression
1		addsubs4d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsubs4d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addsubs4d.3	$$⊢\phi \to C\in \mathrm{No}$$
4		addsubs4d.4	$$⊢\phi \to D\in \mathrm{No}$$
5	1 2 3	addsubsd	$$⊢\phi \to \left(A{+}_{s}B\right){-}_{s}C=\left(A{-}_{s}C\right){+}_{s}B$$
6	5	oveq1d	$$⊢\phi \to \left(\left(A{+}_{s}B\right){-}_{s}C\right){-}_{s}D=\left(\left(A{-}_{s}C\right){+}_{s}B\right){-}_{s}D$$
7	1 2	addscld	$$⊢\phi \to A{+}_{s}B\in \mathrm{No}$$
8	7 3 4	subsubs4d	$$⊢\phi \to \left(\left(A{+}_{s}B\right){-}_{s}C\right){-}_{s}D=\left(A{+}_{s}B\right){-}_s\left(C{+}_{s}D\right)$$
9	1 3	subscld	$$⊢\phi \to A{-}_{s}C\in \mathrm{No}$$
10	9 2 4	addsubsassd	$$⊢\phi \to \left(\left(A{-}_{s}C\right){+}_{s}B\right){-}_{s}D=\left(A{-}_{s}C\right){+}_s\left(B{-}_{s}D\right)$$
11	6 8 10	3eqtr3d	$$⊢\phi \to \left(A{+}_{s}B\right){-}_s\left(C{+}_{s}D\right)=\left(A{-}_{s}C\right){+}_s\left(B{-}_{s}D\right)$$