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Mirrors > Home > MPE Home > Th. List > zfac | Structured version Visualization version Unicode version |
Description: Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 9281. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.) |
Ref | Expression |
---|---|
zfac |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-ac 9281 | . 2 | |
2 | equequ2 1953 | . . . . . . . . . 10 | |
3 | 2 | bibi2d 332 | . . . . . . . . 9 |
4 | elequ2 2004 | . . . . . . . . . . . . 13 | |
5 | 4 | anbi2d 740 | . . . . . . . . . . . 12 |
6 | elequ2 2004 | . . . . . . . . . . . . 13 | |
7 | elequ1 1997 | . . . . . . . . . . . . 13 | |
8 | 6, 7 | anbi12d 747 | . . . . . . . . . . . 12 |
9 | 5, 8 | anbi12d 747 | . . . . . . . . . . 11 |
10 | 9 | cbvexv 2275 | . . . . . . . . . 10 |
11 | 10 | bibi1i 328 | . . . . . . . . 9 |
12 | 3, 11 | syl6bb 276 | . . . . . . . 8 |
13 | 12 | albidv 1849 | . . . . . . 7 |
14 | elequ1 1997 | . . . . . . . . . . . 12 | |
15 | 14 | anbi1d 741 | . . . . . . . . . . 11 |
16 | elequ1 1997 | . . . . . . . . . . . 12 | |
17 | 16 | anbi1d 741 | . . . . . . . . . . 11 |
18 | 15, 17 | anbi12d 747 | . . . . . . . . . 10 |
19 | 18 | exbidv 1850 | . . . . . . . . 9 |
20 | equequ1 1952 | . . . . . . . . 9 | |
21 | 19, 20 | bibi12d 335 | . . . . . . . 8 |
22 | 21 | cbvalv 2273 | . . . . . . 7 |
23 | 13, 22 | syl6bb 276 | . . . . . 6 |
24 | 23 | cbvexv 2275 | . . . . 5 |
25 | 24 | imbi2i 326 | . . . 4 |
26 | 25 | 2albii 1748 | . . 3 |
27 | 26 | exbii 1774 | . 2 |
28 | 1, 27 | mpbi 220 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wex 1704 |
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-11 2034 ax-12 2047 ax-13 2246 ax-ac 9281 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: axacndlem4 9432 |
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