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Mirrors > Home > MPE Home > Th. List > Mathboxes > amosym1 | Structured version Visualization version GIF version |
Description: A symmetry with ∃*.
See negsym1 33322 for more information. (Contributed by Anthony Hart, 13-Sep-2011.) |
Ref | Expression |
---|---|
amosym1 | ⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeu 2603 | . 2 ⊢ (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥)) | |
2 | mofal 33315 | . . . . 5 ⊢ ∃*𝑥⊥ | |
3 | 19.8a 2110 | . . . . . 6 ⊢ (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥) | |
4 | 3 | notnotd 141 | . . . . 5 ⊢ (∃*𝑥⊥ → ¬ ¬ ∃𝑥∃*𝑥⊥) |
5 | 2, 4 | ax-mp 5 | . . . 4 ⊢ ¬ ¬ ∃𝑥∃*𝑥⊥ |
6 | 5 | pm2.21i 117 | . . 3 ⊢ (¬ ∃𝑥∃*𝑥⊥ → ∃*𝑥𝜑) |
7 | 2 | notnoti 140 | . . . . . 6 ⊢ ¬ ¬ ∃*𝑥⊥ |
8 | 7 | nex 1764 | . . . . 5 ⊢ ¬ ∃𝑥 ¬ ∃*𝑥⊥ |
9 | eunex 5139 | . . . . 5 ⊢ (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥) | |
10 | 8, 9 | mto 189 | . . . 4 ⊢ ¬ ∃!𝑥∃*𝑥⊥ |
11 | 10 | pm2.21i 117 | . . 3 ⊢ (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑) |
12 | 6, 11 | ja 175 | . 2 ⊢ ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑) |
13 | 1, 12 | sylbi 209 | 1 ⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊥wfal 1520 ∃wex 1743 ∃*wmo 2546 ∃!weu 2584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-12 2107 ax-nul 5063 ax-pow 5115 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-mo 2548 df-eu 2585 |
This theorem is referenced by: (None) |
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