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Mirrors > Home > ILE Home > Th. List > pm5.18dc | Unicode version |
Description: Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.) |
Ref | Expression |
---|---|
pm5.18dc | DECID DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 776 | . 2 DECID | |
2 | pm5.501 242 | . . . . . . . 8 | |
3 | 2 | a1d 22 | . . . . . . 7 DECID |
4 | 3 | con1biddc 803 | . . . . . 6 DECID |
5 | 4 | imp 122 | . . . . 5 DECID |
6 | pm5.501 242 | . . . . . 6 | |
7 | 6 | adantr 270 | . . . . 5 DECID |
8 | 5, 7 | bitr2d 187 | . . . 4 DECID |
9 | 8 | ex 113 | . . 3 DECID |
10 | dcn 779 | . . . . . . 7 DECID DECID | |
11 | nbn2 645 | . . . . . . . . 9 | |
12 | 11 | a1d 22 | . . . . . . . 8 DECID |
13 | 12 | con1biddc 803 | . . . . . . 7 DECID |
14 | 10, 13 | syl5 32 | . . . . . 6 DECID |
15 | 14 | imp 122 | . . . . 5 DECID |
16 | nbn2 645 | . . . . . 6 | |
17 | 16 | adantr 270 | . . . . 5 DECID |
18 | 15, 17 | bitr2d 187 | . . . 4 DECID |
19 | 18 | ex 113 | . . 3 DECID |
20 | 9, 19 | jaoi 668 | . 2 DECID |
21 | 1, 20 | sylbi 119 | 1 DECID DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 DECID wdc 775 |
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-in1 576 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-dc 776 |
This theorem is referenced by: xor3dc 1318 dfbi3dc 1328 |
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