Theorem int-addassocd

Description: AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-addassocd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	int-addassocd.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	int-addassocd.3	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	int-addassocd.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	int-addassocd	⊢ ( 𝜑 → ( 𝐵 + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		int-addassocd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		int-addassocd.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
3		int-addassocd.3	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
4		int-addassocd.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
5	1	recnd	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
6	2	recnd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
7	3	recnd	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
8	5 6 7	addassd	⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) + 𝐷 ) = ( 𝐴 + ( 𝐶 + 𝐷 ) ) )
9	4	oveq1d	⊢ ( 𝜑 → ( 𝐴 + ( 𝐶 + 𝐷 ) ) = ( 𝐵 + ( 𝐶 + 𝐷 ) ) )
10	8 9	eqtr2d	⊢ ( 𝜑 → ( 𝐵 + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + 𝐷 ) )