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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvdisjdavw2 | Structured version Visualization version GIF version |
Description: Change bound variable and domain in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvdisjdavw2.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷) |
cbvdisjdavw2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvdisjdavw2 | ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvdisjdavw2.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷) | |
2 | 1 | eleq2d 2826 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)) |
3 | cbvdisjdavw2.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) | |
4 | 2, 3 | cbvrmodavw2 36262 | . . 3 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑦 ∈ 𝐵 𝑡 ∈ 𝐷)) |
5 | 4 | albidv 1920 | . 2 ⊢ (𝜑 → (∀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 | 3bitr4g 314 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀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) |
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