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Mirrors > Home > MPE Home > Th. List > reupick2 | Structured version Visualization version GIF version |
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
Ref | Expression |
---|---|
reupick2 | ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancr 547 | . . . . . 6 ⊢ ((𝜓 → 𝜑) → (𝜓 → (𝜑 ∧ 𝜓))) | |
2 | 1 | ralimi 3083 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → ∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓))) |
3 | rexim 3087 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓)) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))) |
5 | reupick3 4319 | . . . . . 6 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓)) | |
6 | 5 | 3exp 1119 | . . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))) |
7 | 6 | com12 32 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃!𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))) |
8 | 4, 7 | syl6 35 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 → (∃!𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))))) |
9 | 8 | 3imp1 1347 | . 2 ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓)) |
10 | rsp 3244 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑))) | |
11 | 10 | 3ad2ant1 1133 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑))) |
12 | 11 | imp 407 | . 2 ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜑)) |
13 | 9, 12 | impbid 211 | 1 ⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ∃!wreu 3374 |
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-10 2137 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-ex 1782 df-nf 1786 df-mo 2534 df-eu 2563 df-ral 3062 df-rex 3071 df-reu 3377 |
This theorem is referenced by: grpoidval 29804 grpoidinv2 29806 grpoinv 29816 |
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