Theorem ixpeq2d

Description: Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Hypotheses	ixpeq2d.1	⊢ Ⅎ 𝑥 𝜑
	ixpeq2d.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion	ixpeq2d	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ixpeq2d.1	⊢ Ⅎ 𝑥 𝜑
2		ixpeq2d.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 ) )
4	1 3	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 )
5		ixpeq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑥 ∈ 𝐴 𝐶 )
6	4 5	syl	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑥 ∈ 𝐴 𝐶 )