Theorem ixpeq12i 37120

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 3084	. . 3 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷
3		ixpeq2 8953	. . 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 37119	. 2 ⊢ X𝑥 ∈ 𝐴 𝐷 = X𝑥 ∈ 𝐵 𝐷
7	4, 6	eqtri 2786	1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷

This proof depends on syntax axioms:   = wceq 1569  ∀wral 3080  Xcixp 8939

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-ss 3901  df-fn 6550  df-ixp 8940

This theorem is used by: (None)