Theorem cbvdisjdavw2 37208

Description: Change bound variable and domain in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbvdisjdavw2.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
cbvdisjdavw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvdisjdavw2	⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷))

Distinct variable groups:   𝜑,𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbvdisjdavw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvdisjdavw2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
2	1	eleq2d 2850	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
3		cbvdisjdavw2.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
4	2, 3	cbvrmodavw2 37202	. . 3 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷))
5	4	albidv 1954	. 2 ⊢ (𝜑 → (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷))
6		df-disj 5074	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶)
7		df-disj 5074	. 2 ⊢ (Disj 𝑦 ∈ 𝐵 𝐷 ↔ ∀𝑡∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷)
8	5, 6, 7	3bitr4g 317	1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2146  ∃*wrmo 3368  Disj wdisj 5073

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-mo 2567  df-cleq 2755  df-clel 2839  df-rmo 3369  df-disj 5074

This theorem is used by: (None)