Theorem ixpeq2dva

Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016)

	Ref	Expression
Hypothesis	ixpeq2dva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion	ixpeq2dva	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ixpeq2dva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
2	1	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 )
3		ixpeq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑥 ∈ 𝐴 𝐶 )
4	2 3	syl	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑥 ∈ 𝐴 𝐶 )