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Metamath Proof Explorer

Theorem addsubsub23

Description: Swap the second and the third terms in a difference of a sum and a difference (or, vice versa, in a sum 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	addsubsub23	⊢ ( 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	addcld	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℂ )
6	5 3 4	subsubd	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) − ( 𝐶 − 𝐷 ) ) = ( ( ( 𝐴 + 𝐵 ) − 𝐶 ) + 𝐷 ) )
7	1 2 3	addsubd	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) − 𝐶 ) = ( ( 𝐴 − 𝐶 ) + 𝐵 ) )
8	7	oveq1d	⊢ ( 𝜑 → ( ( ( 𝐴 + 𝐵 ) − 𝐶 ) + 𝐷 ) = ( ( ( 𝐴 − 𝐶 ) + 𝐵 ) + 𝐷 ) )
9	1 3	subcld	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) ∈ ℂ )
10	9 2 4	addassd	⊢ ( 𝜑 → ( ( ( 𝐴 − 𝐶 ) + 𝐵 ) + 𝐷 ) = ( ( 𝐴 − 𝐶 ) + ( 𝐵 + 𝐷 ) ) )
11	6 8 10	3eqtrd	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) − ( 𝐶 − 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) + ( 𝐵 + 𝐷 ) ) )