Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > reuind | Structured version Visualization version Unicode version |
Description: Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
Ref | Expression |
---|---|
reuind.1 | |
reuind.2 |
Ref | Expression |
---|---|
reuind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuind.2 | . . . . . . . 8 | |
2 | 1 | eleq1d 2686 | . . . . . . 7 |
3 | reuind.1 | . . . . . . 7 | |
4 | 2, 3 | anbi12d 747 | . . . . . 6 |
5 | 4 | cbvexv 2275 | . . . . 5 |
6 | r19.41v 3089 | . . . . . . 7 | |
7 | 6 | exbii 1774 | . . . . . 6 |
8 | rexcom4 3225 | . . . . . 6 | |
9 | risset 3062 | . . . . . . . 8 | |
10 | 9 | anbi1i 731 | . . . . . . 7 |
11 | 10 | exbii 1774 | . . . . . 6 |
12 | 7, 8, 11 | 3bitr4ri 293 | . . . . 5 |
13 | 5, 12 | bitri 264 | . . . 4 |
14 | eqeq2 2633 | . . . . . . . . . 10 | |
15 | 14 | imim2i 16 | . . . . . . . . 9 |
16 | biimpr 210 | . . . . . . . . . . 11 | |
17 | 16 | imim2i 16 | . . . . . . . . . 10 |
18 | an31 841 | . . . . . . . . . . . 12 | |
19 | 18 | imbi1i 339 | . . . . . . . . . . 11 |
20 | impexp 462 | . . . . . . . . . . 11 | |
21 | impexp 462 | . . . . . . . . . . 11 | |
22 | 19, 20, 21 | 3bitr3i 290 | . . . . . . . . . 10 |
23 | 17, 22 | sylib 208 | . . . . . . . . 9 |
24 | 15, 23 | syl 17 | . . . . . . . 8 |
25 | 24 | 2alimi 1740 | . . . . . . 7 |
26 | 19.23v 1902 | . . . . . . . . . 10 | |
27 | an12 838 | . . . . . . . . . . . . . 14 | |
28 | eleq1 2689 | . . . . . . . . . . . . . . . 16 | |
29 | 28 | adantr 481 | . . . . . . . . . . . . . . 15 |
30 | 29 | pm5.32ri 670 | . . . . . . . . . . . . . 14 |
31 | 27, 30 | bitr4i 267 | . . . . . . . . . . . . 13 |
32 | 31 | exbii 1774 | . . . . . . . . . . . 12 |
33 | 19.42v 1918 | . . . . . . . . . . . 12 | |
34 | 32, 33 | bitri 264 | . . . . . . . . . . 11 |
35 | 34 | imbi1i 339 | . . . . . . . . . 10 |
36 | 26, 35 | bitri 264 | . . . . . . . . 9 |
37 | 36 | albii 1747 | . . . . . . . 8 |
38 | 19.21v 1868 | . . . . . . . 8 | |
39 | 37, 38 | bitri 264 | . . . . . . 7 |
40 | 25, 39 | sylib 208 | . . . . . 6 |
41 | 40 | expd 452 | . . . . 5 |
42 | 41 | reximdvai 3015 | . . . 4 |
43 | 13, 42 | syl5bi 232 | . . 3 |
44 | 43 | imp 445 | . 2 |
45 | pm4.24 675 | . . . . . . . . . 10 | |
46 | 45 | biimpi 206 | . . . . . . . . 9 |
47 | prth 595 | . . . . . . . . 9 | |
48 | eqtr3 2643 | . . . . . . . . 9 | |
49 | 46, 47, 48 | syl56 36 | . . . . . . . 8 |
50 | 49 | alanimi 1744 | . . . . . . 7 |
51 | 19.23v 1902 | . . . . . . 7 | |
52 | 50, 51 | sylib 208 | . . . . . 6 |
53 | 52 | com12 32 | . . . . 5 |
54 | 53 | a1d 25 | . . . 4 |
55 | 54 | ralrimivv 2970 | . . 3 |
56 | 55 | adantl 482 | . 2 |
57 | eqeq1 2626 | . . . . 5 | |
58 | 57 | imbi2d 330 | . . . 4 |
59 | 58 | albidv 1849 | . . 3 |
60 | 59 | reu4 3400 | . 2 |
61 | 44, 56, 60 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 wral 2912 wrex 2913 wreu 2914 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-v 3202 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |