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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 2826 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 3 | 2 | anbi1d 631 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶))) |
| 4 | 3 | mobidv 2548 | . . . . 5 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶))) |
| 5 | df-rmo 3379 | . . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶)) | |
| 6 | df-rmo 3379 | . . . . 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 1920 | . . 3 ⊢ (𝜑 → (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)) |
| 9 | df-disj 5109 | . . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶) | |
| 10 | df-disj 5109 | . . 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 5113 | . 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 1538 = wceq 1540 ∈ wcel 2108 ∃*wmo 2537 ∃*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-ral 3061 df-rmo 3379 df-ss 3967 df-disj 5109 |
| This theorem is referenced by: (None) |
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