![]() |
Mathbox for Gino Giotto |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvdisjdavw | Structured version Visualization version GIF version |
Description: Change bound variable in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvdisjdavw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvdisjdavw | ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvdisjdavw.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶) | |
2 | 1 | eleq2d 2823 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶)) |
3 | 2 | cbvrmodavw 36195 | . . 3 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑡 ∈ 𝐶)) |
4 | 3 | albidv 1916 | . 2 ⊢ (𝜑 → (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∀𝑡∃*𝑦 ∈ 𝐴 𝑡 ∈ 𝐶)) |
5 | df-disj 5117 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐵) | |
6 | df-disj 5117 | . 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑦 ∈ 𝐴 𝑡 ∈ 𝐶) | |
7 | 4, 5, 6 | 3bitr4g 314 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1533 = wceq 1535 ∈ wcel 2104 ∃*wrmo 3375 Disj wdisj 5116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-mo 2536 df-cleq 2725 df-clel 2812 df-rmo 3376 df-disj 5117 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |