Theorem ixpeq2dv

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

	Ref	Expression
Hypothesis	ixpeq2dv.1	⊢ ( 𝜑 → 𝐵 = 𝐶 )
Assertion	ixpeq2dv	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ixpeq2dv.1	⊢ ( 𝜑 → 𝐵 = 𝐶 )
2	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3	2	ixpeq2dva	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑥 ∈ 𝐴 𝐶 )