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Mirrors > Home > MPE Home > Th. List > reu6i | Structured version Visualization version GIF version |
Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
reu6i | ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2748 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐵)) | |
2 | 1 | bibi2d 342 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝐵))) |
3 | 2 | ralbidv 3174 | . . 3 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵))) |
4 | 3 | rspcev 3581 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) |
5 | reu6 3684 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ∃!wreu 3351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-reu 3354 |
This theorem is referenced by: eqreu 3687 riota5f 7342 negeu 11391 creur 12147 creui 12148 reuccatpfxs1 14635 lublecl 18250 dfod2 19346 lmieu 27726 esum2dlem 32691 fvineqsneu 35882 poimirlem16 36094 poimirlem17 36095 poimirlem19 36097 poimirlem20 36098 poimirlem22 36100 renegeulemv 40823 sn-subeu 40881 |
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