Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ereq1 | Unicode version |
Description: Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ereq1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releq 4591 | . . 3 | |
2 | dmeq 4709 | . . . 4 | |
3 | 2 | eqeq1d 2126 | . . 3 |
4 | cnveq 4683 | . . . . . 6 | |
5 | coeq1 4666 | . . . . . . 7 | |
6 | coeq2 4667 | . . . . . . 7 | |
7 | 5, 6 | eqtrd 2150 | . . . . . 6 |
8 | 4, 7 | uneq12d 3201 | . . . . 5 |
9 | 8 | sseq1d 3096 | . . . 4 |
10 | sseq2 3091 | . . . 4 | |
11 | 9, 10 | bitrd 187 | . . 3 |
12 | 1, 3, 11 | 3anbi123d 1275 | . 2 |
13 | df-er 6397 | . 2 | |
14 | df-er 6397 | . 2 | |
15 | 12, 13, 14 | 3bitr4g 222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 947 wceq 1316 cun 3039 wss 3041 ccnv 4508 cdm 4509 ccom 4513 wrel 4514 wer 6394 |
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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
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-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-er 6397 |
This theorem is referenced by: riinerm 6470 |
Copyright terms: Public domain | W3C validator |