Theorem ixpeq2d 45073

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

Hypotheses

Ref	Expression
ixpeq2d.1	⊢ Ⅎ𝑥𝜑
ixpeq2d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
ixpeq2d	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Proof of Theorem ixpeq2d

Step	Hyp	Ref	Expression
1		ixpeq2d.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		ixpeq2d.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	ex 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶))
4	1, 3	ralrimi 3257	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
5		ixpeq2 8951	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)
6	4, 5	syl 17	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  ∀wral 3061  Xcixp 8937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-ss 3968  df-ixp 8938

This theorem is referenced by:  hoicvrrex  46571  ovnlecvr  46573  ovnhoilem1  46616  hoi2toco  46622  ovnlecvr2  46625  opnvonmbllem1  46647