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Mirrors > Home > ILE Home > Th. List > xordc1 | Unicode version |
Description: Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
Ref | Expression |
---|---|
xordc1 | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | andir 732 | . . 3 | |
2 | simpl 102 | . . . 4 | |
3 | imnan 624 | . . . . . 6 | |
4 | ancom 253 | . . . . . 6 | |
5 | 3, 4 | xchbinxr 608 | . . . . 5 |
6 | pm3.35 329 | . . . . 5 | |
7 | 5, 6 | sylan2br 272 | . . . 4 |
8 | 2, 7 | orim12i 676 | . . 3 |
9 | 1, 8 | sylbi 114 | . 2 |
10 | df-xor 1267 | . 2 | |
11 | df-dc 743 | . 2 DECID | |
12 | 9, 10, 11 | 3imtr4i 190 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 DECID wdc 742 wxo 1266 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-xor 1267 |
This theorem is referenced by: (None) |
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