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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 3140 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
3 | disjeq2 5110 | . 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Disj wdisj 5106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-mo 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rmo 3370 df-v 3470 df-in 3950 df-ss 3960 df-disj 5107 |
This theorem is referenced by: disjeq12d 5115 iunmbl 25437 uniioovol 25463 tocyccntz 32809 carsggect 33847 voliunnfl 37045 nnfoctbdjlem 45743 meadjiun 45754 |
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