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Mirrors > Home > MPE Home > Th. List > eupicka | Structured version Visualization version GIF version |
Description: Version of eupick 2565 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 2508 | . . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | nfe1 2067 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓) | |
3 | 1, 2 | nfan 1868 | . 2 ⊢ Ⅎ𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) |
4 | eupick 2565 | . 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
5 | 3, 4 | alrimi 2120 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1521 ∃wex 1744 ∃!weu 2498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-10 2059 ax-11 2074 ax-12 2087 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-eu 2502 df-mo 2503 |
This theorem is referenced by: eupickbi 2568 frege124d 38370 sbiota1 38952 |
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