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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eunex | Structured version Visualization version GIF version |
Description: Remove dependency on ax-13 2246 from eunex 4859. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-eunex | ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-dtru 32797 | . . . . 5 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 | |
2 | alim 1738 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∀𝑥𝜑 → ∀𝑥 𝑥 = 𝑦)) | |
3 | 1, 2 | mtoi 190 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ¬ ∀𝑥𝜑) |
4 | 3 | exlimiv 1858 | . . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ¬ ∀𝑥𝜑) |
5 | 4 | adantl 482 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ¬ ∀𝑥𝜑) |
6 | eu3v 2498 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | |
7 | exnal 1754 | . 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
8 | 5, 6, 7 | 3imtr4i 281 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∀wal 1481 ∃wex 1704 ∃!weu 2470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-nul 4789 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-eu 2474 df-mo 2475 |
This theorem is referenced by: (None) |
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