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Mirrors > Home > ILE Home > Th. List > dfif3 | Unicode version |
Description: Alternate definition of the conditional operator df-if 3352. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
dfif3.1 |
Ref | Expression |
---|---|
dfif3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfif6 3353 | . 2 | |
2 | dfif3.1 | . . . . . 6 | |
3 | biidd 170 | . . . . . . 7 | |
4 | 3 | cbvabv 2202 | . . . . . 6 |
5 | 2, 4 | eqtri 2101 | . . . . 5 |
6 | 5 | ineq2i 3164 | . . . 4 |
7 | dfrab3 3240 | . . . 4 | |
8 | 6, 7 | eqtr4i 2104 | . . 3 |
9 | dfrab3 3240 | . . . 4 | |
10 | notab 3234 | . . . . . 6 | |
11 | 5 | difeq2i 3087 | . . . . . 6 |
12 | 10, 11 | eqtr4i 2104 | . . . . 5 |
13 | 12 | ineq2i 3164 | . . . 4 |
14 | 9, 13 | eqtr2i 2102 | . . 3 |
15 | 8, 14 | uneq12i 3124 | . 2 |
16 | 1, 15 | eqtr4i 2104 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wceq 1284 cab 2067 crab 2352 cvv 2601 cdif 2970 cun 2971 cin 2972 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-in1 576 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-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 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-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-if 3352 |
This theorem is referenced by: (None) |
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