Theorem subsubadd23

Description: Swap the second and the third terms in a difference of a difference and a sum. (Contributed by AV, 15-Nov-2025)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	addsub4d.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
Assertion	subsubadd23	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) − ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) − ( 𝐵 + 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		addsub4d.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5	1 2 3	sub32d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) − 𝐶 ) = ( ( 𝐴 − 𝐶 ) − 𝐵 ) )
6	5	oveq1d	⊢ ( 𝜑 → ( ( ( 𝐴 − 𝐵 ) − 𝐶 ) − 𝐷 ) = ( ( ( 𝐴 − 𝐶 ) − 𝐵 ) − 𝐷 ) )
7	1 2	subcld	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ∈ ℂ )
8	7 3 4	subsub4d	⊢ ( 𝜑 → ( ( ( 𝐴 − 𝐵 ) − 𝐶 ) − 𝐷 ) = ( ( 𝐴 − 𝐵 ) − ( 𝐶 + 𝐷 ) ) )
9	1 3	subcld	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) ∈ ℂ )
10	9 2 4	subsub4d	⊢ ( 𝜑 → ( ( ( 𝐴 − 𝐶 ) − 𝐵 ) − 𝐷 ) = ( ( 𝐴 − 𝐶 ) − ( 𝐵 + 𝐷 ) ) )
11	6 8 10	3eqtr3d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) − ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) − ( 𝐵 + 𝐷 ) ) )