Theorem subadd4b

Description: Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	subadd4b.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	subadd4b.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subadd4b.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	subadd4b.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
Assertion	subadd4b	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) + ( 𝐶 − 𝐷 ) ) = ( ( 𝐴 − 𝐷 ) + ( 𝐶 − 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		subadd4b.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		subadd4b.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subadd4b.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		subadd4b.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5	1 2 4 3	subadd4d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) − ( 𝐷 − 𝐶 ) ) = ( ( 𝐴 + 𝐶 ) − ( 𝐵 + 𝐷 ) ) )
6	1 2	subcld	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ∈ ℂ )
7	6 4 3	subsub2d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) − ( 𝐷 − 𝐶 ) ) = ( ( 𝐴 − 𝐵 ) + ( 𝐶 − 𝐷 ) ) )
8	2 4	addcomd	⊢ ( 𝜑 → ( 𝐵 + 𝐷 ) = ( 𝐷 + 𝐵 ) )
9	8	oveq2d	⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) − ( 𝐵 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) − ( 𝐷 + 𝐵 ) ) )
10	1 3 4 2	addsub4d	⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) − ( 𝐷 + 𝐵 ) ) = ( ( 𝐴 − 𝐷 ) + ( 𝐶 − 𝐵 ) ) )
11	9 10	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) − ( 𝐵 + 𝐷 ) ) = ( ( 𝐴 − 𝐷 ) + ( 𝐶 − 𝐵 ) ) )
12	5 7 11	3eqtr3d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) + ( 𝐶 − 𝐷 ) ) = ( ( 𝐴 − 𝐷 ) + ( 𝐶 − 𝐵 ) ) )