Theorem itgeq2dv

Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014)

	Ref	Expression
Hypothesis	itgeq2dv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion	itgeq2dv	⊢ ( 𝜑 → ∫ 𝐴 𝐵 d 𝑥 = ∫ 𝐴 𝐶 d 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		itgeq2dv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
2	1	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 )
3		itgeq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∫ 𝐴 𝐵 d 𝑥 = ∫ 𝐴 𝐶 d 𝑥 )
4	2 3	syl	⊢ ( 𝜑 → ∫ 𝐴 𝐵 d 𝑥 = ∫ 𝐴 𝐶 d 𝑥 )