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Mirrors > Home > MPE Home > Th. List > eupicka | Structured version Visualization version GIF version |
Description: Version of eupick 2524 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 2468 | . . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | nfe1 2014 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓) | |
3 | 1, 2 | nfan 1816 | . 2 ⊢ Ⅎ𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) |
4 | eupick 2524 | . 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
5 | 3, 4 | alrimi 2069 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 ∃!weu 2458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-eu 2462 df-mo 2463 |
This theorem is referenced by: eupickbi 2527 frege124d 37072 sbiota1 37657 |
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