Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > optocl | Unicode version |
Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.) |
Ref | Expression |
---|---|
optocl.1 | |
optocl.2 | |
optocl.3 |
Ref | Expression |
---|---|
optocl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp3 4563 | . . 3 | |
2 | opelxp 4539 | . . . . . . 7 | |
3 | optocl.3 | . . . . . . 7 | |
4 | 2, 3 | sylbi 120 | . . . . . 6 |
5 | optocl.2 | . . . . . 6 | |
6 | 4, 5 | syl5ib 153 | . . . . 5 |
7 | 6 | imp 123 | . . . 4 |
8 | 7 | exlimivv 1852 | . . 3 |
9 | 1, 8 | sylbi 120 | . 2 |
10 | optocl.1 | . 2 | |
11 | 9, 10 | eleq2s 2212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cop 3500 cxp 4507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-xp 4515 |
This theorem is referenced by: 2optocl 4586 3optocl 4587 ecoptocl 6484 ax1rid 7653 ax0id 7654 axcnre 7657 |
Copyright terms: Public domain | W3C validator |