Theorem ixpeq1d

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

	Ref	Expression
Hypothesis	ixpeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	ixpeq1d	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐶 = X 𝑥 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ixpeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		ixpeq1	⊢ ( 𝐴 = 𝐵 → X 𝑥 ∈ 𝐴 𝐶 = X 𝑥 ∈ 𝐵 𝐶 )
3	1 2	syl	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐶 = X 𝑥 ∈ 𝐵 𝐶 )