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| Mirrors > Home > MPE Home > Th. List > disjeq2dv | Structured version Visualization version GIF version | ||
| Description: Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) | 
| Ref | Expression | 
|---|---|
| disjeq2dv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| disjeq2dv | ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | disjeq2dv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
| 2 | 1 | ralrimiva 3145 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) | 
| 3 | disjeq2 5113 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) | |
| 4 | 2, 3 | syl 17 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 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: disjeq12d 5118 iunmbl 25589 uniioovol 25615 tocyccntz 33165 carsggect 34321 disjeq12dv 36217 voliunnfl 37672 nnfoctbdjlem 46475 meadjiun 46486 | 
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