Theorem subsubs2d

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

	Ref	Expression
Hypotheses	subsubs4d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	subsubs4d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	subsubs4d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	subsubs2d	⊢ ( 𝜑 → ( 𝐴 -s ( 𝐵 -s 𝐶 ) ) = ( 𝐴 +s ( 𝐶 -s 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		subsubs4d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		subsubs4d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		subsubs4d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4	2 3	subscld	⊢ ( 𝜑 → ( 𝐵 -s 𝐶 ) ∈ No )
5	1 4	subsvald	⊢ ( 𝜑 → ( 𝐴 -s ( 𝐵 -s 𝐶 ) ) = ( 𝐴 +s ( -us ‘ ( 𝐵 -s 𝐶 ) ) ) )
6	2 3	negsubsdi2d	⊢ ( 𝜑 → ( -us ‘ ( 𝐵 -s 𝐶 ) ) = ( 𝐶 -s 𝐵 ) )
7	6	oveq2d	⊢ ( 𝜑 → ( 𝐴 +s ( -us ‘ ( 𝐵 -s 𝐶 ) ) ) = ( 𝐴 +s ( 𝐶 -s 𝐵 ) ) )
8	5 7	eqtrd	⊢ ( 𝜑 → ( 𝐴 -s ( 𝐵 -s 𝐶 ) ) = ( 𝐴 +s ( 𝐶 -s 𝐵 ) ) )