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Mirrors > Home > MPE Home > Th. List > eupicka | Structured version Visualization version GIF version |
Description: Version of eupick 2623 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 2576 | . . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | nfe1 2139 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓) | |
3 | 1, 2 | nfan 1894 | . 2 ⊢ Ⅎ𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) |
4 | eupick 2623 | . 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
5 | 3, 4 | alrimi 2198 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1531 ∃wex 1773 ∃!weu 2556 |
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-10 2129 ax-11 2146 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-mo 2528 df-eu 2557 |
This theorem is referenced by: eupickbi 2626 frege124d 43070 sbiota1 43751 |
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