Theorem int-eqmvtd

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

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

Proof

Step	Hyp	Ref	Expression
1		int-eqmvtd.1	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
2		int-eqmvtd.2	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
3		int-eqmvtd.3	⊢ ( 𝜑 → 𝐴 = 𝐵 )
4		int-eqmvtd.4	⊢ ( 𝜑 → 𝐴 = ( 𝐶 + 𝐷 ) )
5	3 4	eqtr3d	⊢ ( 𝜑 → 𝐵 = ( 𝐶 + 𝐷 ) )
6	5	oveq1d	⊢ ( 𝜑 → ( 𝐵 − 𝐷 ) = ( ( 𝐶 + 𝐷 ) − 𝐷 ) )
7	1	recnd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
8	2	recnd	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
9	7 8	pncand	⊢ ( 𝜑 → ( ( 𝐶 + 𝐷 ) − 𝐷 ) = 𝐶 )
10	6 9	eqtrd	⊢ ( 𝜑 → ( 𝐵 − 𝐷 ) = 𝐶 )
11	10	eqcomd	⊢ ( 𝜑 → 𝐶 = ( 𝐵 − 𝐷 ) )