New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > opelopabsb | Unicode version |
Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
opelopabsb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4640 | . . 3 | |
2 | brex 4689 | . . 3 | |
3 | 1, 2 | sylbir 204 | . 2 |
4 | sbcex 3055 | . . 3 | |
5 | spesbc 3127 | . . . 4 | |
6 | sbcex 3055 | . . . . 5 | |
7 | 6 | exlimiv 1634 | . . . 4 |
8 | 5, 7 | syl 15 | . . 3 |
9 | 4, 8 | jca 518 | . 2 |
10 | opeq1 4578 | . . . . 5 | |
11 | 10 | eleq1d 2419 | . . . 4 |
12 | dfsbcq2 3049 | . . . 4 | |
13 | 11, 12 | bibi12d 312 | . . 3 |
14 | opeq2 4579 | . . . . 5 | |
15 | 14 | eleq1d 2419 | . . . 4 |
16 | dfsbcq2 3049 | . . . . 5 | |
17 | 16 | sbcbidv 3100 | . . . 4 |
18 | 15, 17 | bibi12d 312 | . . 3 |
19 | nfopab1 4628 | . . . . . 6 | |
20 | 19 | nfel2 2501 | . . . . 5 |
21 | nfs1v 2106 | . . . . 5 | |
22 | 20, 21 | nfbi 1834 | . . . 4 |
23 | opeq1 4578 | . . . . . 6 | |
24 | 23 | eleq1d 2419 | . . . . 5 |
25 | sbequ12 1919 | . . . . 5 | |
26 | 24, 25 | bibi12d 312 | . . . 4 |
27 | nfopab2 4629 | . . . . . . 7 | |
28 | 27 | nfel2 2501 | . . . . . 6 |
29 | nfs1v 2106 | . . . . . 6 | |
30 | 28, 29 | nfbi 1834 | . . . . 5 |
31 | opeq2 4579 | . . . . . . 7 | |
32 | 31 | eleq1d 2419 | . . . . . 6 |
33 | sbequ12 1919 | . . . . . 6 | |
34 | 32, 33 | bibi12d 312 | . . . . 5 |
35 | opabid 4695 | . . . . 5 | |
36 | 30, 34, 35 | chvar 1986 | . . . 4 |
37 | 22, 26, 36 | chvar 1986 | . . 3 |
38 | 13, 18, 37 | vtocl2g 2918 | . 2 |
39 | 3, 9, 38 | pm5.21nii 342 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 wex 1541 wceq 1642 wsb 1648 wcel 1710 cvv 2859 wsbc 3046 cop 4561 copab 4622 class class class wbr 4639 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 |
This theorem is referenced by: brabsb 4698 opelopabaf 4710 opelopabf 4711 inopab 4862 cnvopab 5030 |
Copyright terms: Public domain | W3C validator |