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| Mirrors > Home > MPE Home > Th. List > eupicka | Structured version Visualization version GIF version | ||
| Description: Version of eupick 2633 with closed formulas. (Contributed by NM, 6-Sep-2008.) | 
| Ref | Expression | 
|---|---|
| eupicka | ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfeu1 2588 | . . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
| 2 | nfe1 2150 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓) | |
| 3 | 1, 2 | nfan 1899 | . 2 ⊢ Ⅎ𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) | 
| 4 | eupick 2633 | . 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
| 5 | 3, 4 | alrimi 2213 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∃wex 1779 ∃!weu 2568 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: eupickbi 2636 frege124d 43774 sbiota1 44453 | 
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