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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 2823 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)) |
4 | 1, 3 | cbvrmovw2 36171 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷) |
5 | 4 | albii 1814 | . 2 ⊢ (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷) |
6 | df-disj 5117 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶) | |
7 | df-disj 5117 | . 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 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) |
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