Theorem ixpeq12dv 36163

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

Hypotheses

Ref	Expression
ixpeq12dv.1	⊢ (𝜑 → 𝐴 = 𝐵)
ixpeq12dv.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
ixpeq12dv	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷)

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem ixpeq12dv

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ixpeq12dv.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	eleq2d 2819	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	abbidv 2800	. . . . . 6 ⊢ (𝜑 → {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ 𝑥 ∈ 𝐵})
4	3	fneq2d 6629	. . . . 5 ⊢ (𝜑 → (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐵}))
5	2	imbi1d 341	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → (𝑡‘𝑥) ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → (𝑡‘𝑥) ∈ 𝐶)))
6	5	albidv 1919	. . . . . 6 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → (𝑡‘𝑥) ∈ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝑡‘𝑥) ∈ 𝐶)))
7		df-ral 3051	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑡‘𝑥) ∈ 𝐶))
8		df-ral 3051	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 (𝑡‘𝑥) ∈ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝑡‘𝑥) ∈ 𝐶))
9	6, 7, 8	3bitr4g 314	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝑡‘𝑥) ∈ 𝐶))
10	4, 9	anbi12d 632	. . . 4 ⊢ (𝜑 → ((𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶) ↔ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑡‘𝑥) ∈ 𝐶)))
11	10	abbidv 2800	. . 3 ⊢ (𝜑 → {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)} = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑡‘𝑥) ∈ 𝐶)})
12		df-ixp 8907	. . 3 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)}
13		df-ixp 8907	. . 3 ⊢ X𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑡‘𝑥) ∈ 𝐶)}
14	11, 12, 13	3eqtr4g 2794	. 2 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)
15		ixpeq12dv.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
16	15	ixpeq2dv 8922	. 2 ⊢ (𝜑 → X𝑥 ∈ 𝐵 𝐶 = X𝑥 ∈ 𝐵 𝐷)
17	14, 16	eqtrd 2769	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2107  {cab 2712  ∀wral 3050   Fn wfn 6523  ‘cfv 6528  Xcixp 8906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-ss 3941  df-fn 6531  df-ixp 8907

This theorem is referenced by: (None)