Theorem subaddsd

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

	Ref	Expression
Hypotheses	subaddsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	subaddsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	subaddsd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	subaddsd	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) = 𝐶 ↔ ( 𝐵 +s 𝐶 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		subaddsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		subaddsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		subaddsd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		subadds	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 -s 𝐵 ) = 𝐶 ↔ ( 𝐵 +s 𝐶 ) = 𝐴 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) = 𝐶 ↔ ( 𝐵 +s 𝐶 ) = 𝐴 ) )