Theorem cbvixpdavw 36678

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 2852	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	2	cbvabdavw 36656	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴})
4	3	fneq2d 6630	. . . 4 ⊢ (𝜑 → (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐴}))
5		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
6	5	fveq2d 6886	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡‘𝑥) = (𝑡‘𝑦))
7		cbvixpdavw.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
8	6, 7	eleq12d 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑡‘𝑥) ∈ 𝐵 ↔ (𝑡‘𝑦) ∈ 𝐶))
9	8	cbvraldva 3251	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑡‘𝑦) ∈ 𝐶))
10	4, 9	anbi12d 643	. . 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 2835	. 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 8895	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐵)}
13		df-ixp 8895	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐴} ∧ ∀𝑦 ∈ 𝐴 (𝑡‘𝑦) ∈ 𝐶)}
14	11, 12, 13	3eqtr4g 2829	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085   Fn wfn 6532  ‘cfv 6537  Xcixp 8894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fn 6540  df-fv 6545  df-ixp 8895

This theorem is referenced by: (None)