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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjeq12dv | Structured version Visualization version GIF version |
Description: Equality theorem for disjoint collection. Deduction version. (Contributed by GG, 1-Sep-2025.) |
Ref | Expression |
---|---|
disjeq12dv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
disjeq12dv.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
disjeq12dv | ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjeq12dv.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | eleq2d 2823 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
3 | 2 | anbi1d 630 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶))) |
4 | 3 | mobidv 2545 | . . . . 5 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶))) |
5 | df-rmo 3376 | . . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶)) | |
6 | df-rmo 3376 | . . . . 5 ⊢ (∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶)) | |
7 | 4, 5, 6 | 3bitr4g 314 | . . . 4 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)) |
8 | 7 | albidv 1916 | . . 3 ⊢ (𝜑 → (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)) |
9 | df-disj 5117 | . . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶) | |
10 | df-disj 5117 | . . 3 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶) | |
11 | 8, 9, 10 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) |
12 | disjeq12dv.2 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) | |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
14 | 13 | disjeq2dv 5121 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
15 | 11, 14 | bitrd 279 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1533 = wceq 1535 ∈ wcel 2104 ∃*wmo 2534 ∃*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-ral 3058 df-rmo 3376 df-ss 3980 df-disj 5117 |
This theorem is referenced by: (None) |
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