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Mirrors > Home > ILE Home > Th. List > ifandc | Unicode version |
Description: Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
ifandc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 805 | . 2 DECID | |
2 | iftrue 3449 | . . . 4 | |
3 | ibar 299 | . . . . 5 | |
4 | 3 | ifbid 3463 | . . . 4 |
5 | 2, 4 | eqtr2d 2151 | . . 3 |
6 | simpl 108 | . . . . . 6 | |
7 | 6 | con3i 606 | . . . . 5 |
8 | 7 | iffalsed 3454 | . . . 4 |
9 | iffalse 3452 | . . . 4 | |
10 | 8, 9 | eqtr4d 2153 | . . 3 |
11 | 5, 10 | jaoi 690 | . 2 |
12 | 1, 11 | sylbi 120 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 682 DECID wdc 804 wceq 1316 cif 3444 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 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-dc 805 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-if 3445 |
This theorem is referenced by: isumss 11115 |
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