Theorem cbvdisjvw2 36463

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

Hypotheses

Ref	Expression
cbvdisjvw2.1	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
cbvdisjvw2.2	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvdisjvw2	⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷)

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

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

Proof of Theorem cbvdisjvw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvdisjvw2.2	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
2		cbvdisjvw2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
3	2	eleq2d 2825	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
4	1, 3	cbvrmovw2 36456	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷)
5	4	albii 1826	. 2 ⊢ (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷)
6		df-disj 5040	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶)
7		df-disj 5040	. 2 ⊢ (Disj 𝑦 ∈ 𝐵 𝐷 ↔ ∀𝑡∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷)
8	5, 6, 7	3bitr4i 304	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545   = wceq 1547   ∈ wcel 2119  ∃*wrmo 3343  Disj wdisj 5039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-cleq 2731  df-clel 2814  df-rmo 3344  df-disj 5040

This theorem is referenced by: (None)