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Theorem ixpeq12dv 36159
Description: Equality theorem for infinite Cartesian product. Deduction version. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
ixpeq12dv.1 (𝜑𝐴 = 𝐵)
ixpeq12dv.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
ixpeq12dv (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 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.
StepHypRef Expression
1 ixpeq12dv.1 . . . . . . . 8 (𝜑𝐴 = 𝐵)
21eleq2d 2823 . . . . . . 7 (𝜑 → (𝑥𝐴𝑥𝐵))
32abbidv 2804 . . . . . 6 (𝜑 → {𝑥𝑥𝐴} = {𝑥𝑥𝐵})
43fneq2d 6658 . . . . 5 (𝜑 → (𝑡 Fn {𝑥𝑥𝐴} ↔ 𝑡 Fn {𝑥𝑥𝐵}))
52imbi1d 341 . . . . . . 7 (𝜑 → ((𝑥𝐴 → (𝑡𝑥) ∈ 𝐶) ↔ (𝑥𝐵 → (𝑡𝑥) ∈ 𝐶)))
65albidv 1916 . . . . . 6 (𝜑 → (∀𝑥(𝑥𝐴 → (𝑡𝑥) ∈ 𝐶) ↔ ∀𝑥(𝑥𝐵 → (𝑡𝑥) ∈ 𝐶)))
7 df-ral 3058 . . . . . 6 (∀𝑥𝐴 (𝑡𝑥) ∈ 𝐶 ↔ ∀𝑥(𝑥𝐴 → (𝑡𝑥) ∈ 𝐶))
8 df-ral 3058 . . . . . 6 (∀𝑥𝐵 (𝑡𝑥) ∈ 𝐶 ↔ ∀𝑥(𝑥𝐵 → (𝑡𝑥) ∈ 𝐶))
96, 7, 83bitr4g 314 . . . . 5 (𝜑 → (∀𝑥𝐴 (𝑡𝑥) ∈ 𝐶 ↔ ∀𝑥𝐵 (𝑡𝑥) ∈ 𝐶))
104, 9anbi12d 631 . . . 4 (𝜑 → ((𝑡 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑡𝑥) ∈ 𝐶) ↔ (𝑡 Fn {𝑥𝑥𝐵} ∧ ∀𝑥𝐵 (𝑡𝑥) ∈ 𝐶)))
1110abbidv 2804 . . 3 (𝜑 → {𝑡 ∣ (𝑡 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑡𝑥) ∈ 𝐶)} = {𝑡 ∣ (𝑡 Fn {𝑥𝑥𝐵} ∧ ∀𝑥𝐵 (𝑡𝑥) ∈ 𝐶)})
12 df-ixp 8931 . . 3 X𝑥𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑡𝑥) ∈ 𝐶)}
13 df-ixp 8931 . . 3 X𝑥𝐵 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥𝑥𝐵} ∧ ∀𝑥𝐵 (𝑡𝑥) ∈ 𝐶)}
1411, 12, 133eqtr4g 2798 . 2 (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
15 ixpeq12dv.2 . . 3 (𝜑𝐶 = 𝐷)
1615ixpeq2dv 8946 . 2 (𝜑X𝑥𝐵 𝐶 = X𝑥𝐵 𝐷)
1714, 16eqtrd 2773 1 (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1533   = wceq 1535  wcel 2104  {cab 2710  wral 3057   Fn wfn 6553  cfv 6558  Xcixp 8930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-ss 3980  df-fn 6561  df-ixp 8931
This theorem is referenced by: (None)
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