Theorem cbvixpdavw 36320

Description: Change bound variable in an indexed Cartesian product. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbvixpdavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvixpdavw	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		eleq1w 2814	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	2	cbvabdavw 36298	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴})
4	3	fneq2d 6575	. . . 4 ⊢ (𝜑 → (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐴}))
5		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
6	5	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡‘𝑥) = (𝑡‘𝑦))
7		cbvixpdavw.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
8	6, 7	eleq12d 2825	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑡‘𝑥) ∈ 𝐵 ↔ (𝑡‘𝑦) ∈ 𝐶))
9	8	cbvraldva 3212	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑡‘𝑦) ∈ 𝐶))
10	4, 9	anbi12d 632	. . 3 ⊢ (𝜑 → ((𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐵) ↔ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐴} ∧ ∀𝑦 ∈ 𝐴 (𝑡‘𝑦) ∈ 𝐶)))
11	10	abbidv 2797	. 2 ⊢ (𝜑 → {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐵)} = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐴} ∧ ∀𝑦 ∈ 𝐴 (𝑡‘𝑦) ∈ 𝐶)})
12		df-ixp 8822	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐵)}
13		df-ixp 8822	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐴} ∧ ∀𝑦 ∈ 𝐴 (𝑡‘𝑦) ∈ 𝐶)}
14	11, 12, 13	3eqtr4g 2791	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047   Fn wfn 6476  ‘cfv 6481  Xcixp 8821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fn 6484  df-fv 6489  df-ixp 8822

This theorem is referenced by: (None)