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Mirrors > Home > MPE Home > Th. List > opabex3d | Structured version Visualization version Unicode version |
Description: Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.) |
Ref | Expression |
---|---|
opabex3d.1 | |
opabex3d.2 |
Ref | Expression |
---|---|
opabex3d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42v 1918 | . . . . . 6 | |
2 | an12 838 | . . . . . . 7 | |
3 | 2 | exbii 1774 | . . . . . 6 |
4 | elxp 5131 | . . . . . . . 8 | |
5 | excom 2042 | . . . . . . . . 9 | |
6 | an12 838 | . . . . . . . . . . . . 13 | |
7 | velsn 4193 | . . . . . . . . . . . . . 14 | |
8 | 7 | anbi1i 731 | . . . . . . . . . . . . 13 |
9 | 6, 8 | bitri 264 | . . . . . . . . . . . 12 |
10 | 9 | exbii 1774 | . . . . . . . . . . 11 |
11 | vex 3203 | . . . . . . . . . . . 12 | |
12 | opeq1 4402 | . . . . . . . . . . . . . 14 | |
13 | 12 | eqeq2d 2632 | . . . . . . . . . . . . 13 |
14 | 13 | anbi1d 741 | . . . . . . . . . . . 12 |
15 | 11, 14 | ceqsexv 3242 | . . . . . . . . . . 11 |
16 | 10, 15 | bitri 264 | . . . . . . . . . 10 |
17 | 16 | exbii 1774 | . . . . . . . . 9 |
18 | 5, 17 | bitri 264 | . . . . . . . 8 |
19 | nfv 1843 | . . . . . . . . . 10 | |
20 | nfsab1 2612 | . . . . . . . . . 10 | |
21 | 19, 20 | nfan 1828 | . . . . . . . . 9 |
22 | nfv 1843 | . . . . . . . . 9 | |
23 | opeq2 4403 | . . . . . . . . . . 11 | |
24 | 23 | eqeq2d 2632 | . . . . . . . . . 10 |
25 | sbequ12 2111 | . . . . . . . . . . . 12 | |
26 | 25 | equcoms 1947 | . . . . . . . . . . 11 |
27 | df-clab 2609 | . . . . . . . . . . 11 | |
28 | 26, 27 | syl6rbbr 279 | . . . . . . . . . 10 |
29 | 24, 28 | anbi12d 747 | . . . . . . . . 9 |
30 | 21, 22, 29 | cbvex 2272 | . . . . . . . 8 |
31 | 4, 18, 30 | 3bitri 286 | . . . . . . 7 |
32 | 31 | anbi2i 730 | . . . . . 6 |
33 | 1, 3, 32 | 3bitr4ri 293 | . . . . 5 |
34 | 33 | exbii 1774 | . . . 4 |
35 | eliun 4524 | . . . . 5 | |
36 | df-rex 2918 | . . . . 5 | |
37 | 35, 36 | bitri 264 | . . . 4 |
38 | elopab 4983 | . . . 4 | |
39 | 34, 37, 38 | 3bitr4i 292 | . . 3 |
40 | 39 | eqriv 2619 | . 2 |
41 | opabex3d.1 | . . 3 | |
42 | snex 4908 | . . . . 5 | |
43 | opabex3d.2 | . . . . 5 | |
44 | xpexg 6960 | . . . . 5 | |
45 | 42, 43, 44 | sylancr 695 | . . . 4 |
46 | 45 | ralrimiva 2966 | . . 3 |
47 | iunexg 7143 | . . 3 | |
48 | 41, 46, 47 | syl2anc 693 | . 2 |
49 | 40, 48 | syl5eqelr 2706 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wex 1704 wsb 1880 wcel 1990 cab 2608 wral 2912 wrex 2913 cvv 3200 csn 4177 cop 4183 ciun 4520 copab 4712 cxp 5112 |
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-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 |
This theorem is referenced by: wksfval 26505 fpwrelmap 29508 cnvepresex 34104 opabresex0d 41304 upwlksfval 41716 |
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