|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > disjeq12d | Structured version Visualization version GIF version | ||
| Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) | 
| Ref | Expression | 
|---|---|
| disjeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| disjeq12d.1 | ⊢ (𝜑 → 𝐶 = 𝐷) | 
| Ref | Expression | 
|---|---|
| disjeq12d | ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | disjeq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | 1 | disjeq1d 5117 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) | 
| 3 | disjeq12d.1 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 4 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) | 
| 5 | 4 | disjeq2dv 5114 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) | 
| 6 | 2, 5 | bitrd 279 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 Disj wdisj 5109 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-mo 2539 df-cleq 2728 df-clel 2815 df-ral 3061 df-rmo 3379 df-ss 3967 df-disj 5110 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |