Theorem ixpeq12i 36636

Description: Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
ixpeq12i.1	⊢ 𝐴 = 𝐵
ixpeq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
ixpeq12i	⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷

Proof of Theorem ixpeq12i

Step	Hyp	Ref	Expression
1		ixpeq12i.2	. . . 4 ⊢ 𝐶 = 𝐷
2	1	rgenw 3089	. . 3 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷
3		ixpeq2 8909	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐴 𝐷)
4	2, 3	ax-mp 5	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐴 𝐷
5		ixpeq12i.1	. . 3 ⊢ 𝐴 = 𝐵
6	5	ixpeq1i 36635	. 2 ⊢ X𝑥 ∈ 𝐴 𝐷 = X𝑥 ∈ 𝐵 𝐷
7	4, 6	eqtri 2792	1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷

Syntax hints:   = wceq 1567  ∀wral 3085  Xcixp 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-ss 3930  df-fn 6540  df-ixp 8896

This theorem is referenced by: (None)