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Mirrors > Home > ILE Home > Th. List > ifcldcd | Unicode version |
Description: Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
Ref | Expression |
---|---|
ifcldcd.a | |
ifcldcd.b | |
ifcldcd.dc | DECID |
Ref | Expression |
---|---|
ifcldcd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 3356 | . . . 4 | |
2 | 1 | adantl 271 | . . 3 |
3 | ifcldcd.a | . . . 4 | |
4 | 3 | adantr 270 | . . 3 |
5 | 2, 4 | eqeltrd 2155 | . 2 |
6 | iffalse 3359 | . . . 4 | |
7 | 6 | adantl 271 | . . 3 |
8 | ifcldcd.b | . . . 4 | |
9 | 8 | adantr 270 | . . 3 |
10 | 7, 9 | eqeltrd 2155 | . 2 |
11 | ifcldcd.dc | . . 3 DECID | |
12 | df-dc 776 | . . 3 DECID | |
13 | 11, 12 | sylib 120 | . 2 |
14 | 5, 10, 13 | mpjaodan 744 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wo 661 DECID wdc 775 wceq 1284 wcel 1433 cif 3351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-if 3352 |
This theorem is referenced by: uzin2 9873 |
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