| Mathbox for Gino Giotto |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvdisjvw2 | Structured version Visualization version GIF version | ||
| Description: Change bound variable and domain in a disjoint collection, using implicit substitution. (Contributed by GG, 14-Aug-2025.) |
| Ref | Expression |
|---|---|
| cbvdisjvw2.1 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
| cbvdisjvw2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| cbvdisjvw2 | ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvdisjvw2.2 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
| 2 | cbvdisjvw2.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
| 3 | 2 | eleq2d 2826 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)) |
| 4 | 1, 3 | cbvrmovw2 36207 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷) |
| 5 | 4 | albii 1819 | . 2 ⊢ (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷) |
| 6 | df-disj 5109 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶) | |
| 7 | df-disj 5109 | . 2 ⊢ (Disj 𝑦 ∈ 𝐵 𝐷 ↔ ∀𝑡∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷) | |
| 8 | 5, 6, 7 | 3bitr4i 303 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃*wrmo 3378 Disj wdisj 5108 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-mo 2539 df-cleq 2728 df-clel 2815 df-rmo 3379 df-disj 5109 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |