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Mirrors > Home > ILE Home > Th. List > eupicka | GIF version |
Description: Version of eupick 2115 with closed formulas. (Contributed by NM, 6-Sep-2008.) |
Ref | Expression |
---|---|
eupicka | ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbeu1 2046 | . . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) | |
2 | hbe1 1505 | . . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∀𝑥∃𝑥(𝜑 ∧ 𝜓)) | |
3 | 1, 2 | hban 1557 | . 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓))) |
4 | eupick 2115 | . 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
5 | 3, 4 | alrimih 1479 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1361 ∃wex 1502 ∃!weu 2036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 |
This theorem is referenced by: eupickbi 2118 |
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