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Theorem cbvixpdavw 36221
Description: Change bound variable in an indexed Cartesian product. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbvixpdavw.1 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
Assertion
Ref Expression
cbvixpdavw (𝜑X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶)
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvixpdavw
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2820 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21adantl 481 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐴))
32cbvabdavw 36199 . . . . 5 (𝜑 → {𝑥𝑥𝐴} = {𝑦𝑦𝐴})
43fneq2d 6658 . . . 4 (𝜑 → (𝑡 Fn {𝑥𝑥𝐴} ↔ 𝑡 Fn {𝑦𝑦𝐴}))
5 simpr 484 . . . . . . 7 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
65fveq2d 6905 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝑡𝑥) = (𝑡𝑦))
7 cbvixpdavw.1 . . . . . 6 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
86, 7eleq12d 2831 . . . . 5 ((𝜑𝑥 = 𝑦) → ((𝑡𝑥) ∈ 𝐵 ↔ (𝑡𝑦) ∈ 𝐶))
98cbvraldva 3235 . . . 4 (𝜑 → (∀𝑥𝐴 (𝑡𝑥) ∈ 𝐵 ↔ ∀𝑦𝐴 (𝑡𝑦) ∈ 𝐶))
104, 9anbi12d 631 . . 3 (𝜑 → ((𝑡 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑡𝑥) ∈ 𝐵) ↔ (𝑡 Fn {𝑦𝑦𝐴} ∧ ∀𝑦𝐴 (𝑡𝑦) ∈ 𝐶)))
1110abbidv 2804 . 2 (𝜑 → {𝑡 ∣ (𝑡 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑡𝑥) ∈ 𝐵)} = {𝑡 ∣ (𝑡 Fn {𝑦𝑦𝐴} ∧ ∀𝑦𝐴 (𝑡𝑦) ∈ 𝐶)})
12 df-ixp 8931 . 2 X𝑥𝐴 𝐵 = {𝑡 ∣ (𝑡 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑡𝑥) ∈ 𝐵)}
13 df-ixp 8931 . 2 X𝑦𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑦𝑦𝐴} ∧ ∀𝑦𝐴 (𝑡𝑦) ∈ 𝐶)}
1411, 12, 133eqtr4g 2798 1 (𝜑X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = 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-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-iota 6510  df-fn 6561  df-fv 6566  df-ixp 8931
This theorem is referenced by: (None)
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