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Mirrors > Home > MPE Home > Th. List > eupickbi | Structured version Visualization version GIF version |
Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Ref | Expression |
---|---|
eupickbi | ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupicka 2712 | . . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) | |
2 | 1 | ex 413 | . 2 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → ∀𝑥(𝜑 → 𝜓))) |
3 | euex 2655 | . . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | |
4 | exintr 1884 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) | |
5 | 3, 4 | syl5com 31 | . 2 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) |
6 | 2, 5 | impbid 213 | 1 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 ∃!weu 2646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 |
This theorem is referenced by: sbaniota 40644 |
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