Theorem cbvixpdavw 37067

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 2844	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	2	cbvabdavw 37045	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴})
4	3	fneq2d 6633	. . . 4 ⊢ (𝜑 → (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐴}))
5		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
6	5	fveq2d 6889	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡‘𝑥) = (𝑡‘𝑦))
7		cbvixpdavw.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
8	6, 7	eleq12d 2855	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑡‘𝑥) ∈ 𝐵 ↔ (𝑡‘𝑦) ∈ 𝐶))
9	8	cbvraldva 3243	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑡‘𝑦) ∈ 𝐶))
10	4, 9	anbi12d 644	. . 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 2827	. 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 8926	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐵)}
13		df-ixp 8926	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐴} ∧ ∀𝑦 ∈ 𝐴 (𝑡‘𝑦) ∈ 𝐶)}
14	11, 12, 13	3eqtr4g 2821	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077   Fn wfn 6533  ‘cfv 6538  Xcixp 8925

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fn 6541  df-fv 6546  df-ixp 8926

This theorem is used by: (None)