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Mirrors > Home > ILE Home > Th. List > ifsbdc | Unicode version |
Description: Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.) |
Ref | Expression |
---|---|
ifsbdc.1 | |
ifsbdc.2 |
Ref | Expression |
---|---|
ifsbdc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmiddc 826 | . 2 DECID | |
2 | iftrue 3525 | . . . . 5 | |
3 | ifsbdc.1 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | iftrue 3525 | . . . 4 | |
6 | 4, 5 | eqtr4d 2201 | . . 3 |
7 | iffalse 3528 | . . . . 5 | |
8 | ifsbdc.2 | . . . . 5 | |
9 | 7, 8 | syl 14 | . . . 4 |
10 | iffalse 3528 | . . . 4 | |
11 | 9, 10 | eqtr4d 2201 | . . 3 |
12 | 6, 11 | jaoi 706 | . 2 |
13 | 1, 12 | syl 14 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 698 DECID wdc 824 wceq 1343 cif 3520 |
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-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-11 1494 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-if 3521 |
This theorem is referenced by: fvifdc 5508 lgsneg 13565 lgsdilem 13568 |
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