Theorem subaddsd

Description: Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 5-Feb-2025)

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

Proof

Step	Hyp	Ref	Expression
1		subaddsd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		subaddsd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		subaddsd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		subadds	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to \left(A{-}_{s}B=C\leftrightarrow B{+}_{s}C=A\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A{-}_{s}B=C\leftrightarrow B{+}_{s}C=A\right)$$