Theorem assraddsubd

Description: Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 15-Oct-2018)

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

Proof

Step	Hyp	Ref	Expression
1		assraddsubd.1	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
2		assraddsubd.2	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
3		assraddsubd.3	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
4		assraddsubd.4	⊢ ( 𝜑 → 𝐴 = ( ( 𝐵 + 𝐶 ) − 𝐷 ) )
5	1 2 3	addsubassd	⊢ ( 𝜑 → ( ( 𝐵 + 𝐶 ) − 𝐷 ) = ( 𝐵 + ( 𝐶 − 𝐷 ) ) )
6	4 5	eqtrd	⊢ ( 𝜑 → 𝐴 = ( 𝐵 + ( 𝐶 − 𝐷 ) ) )