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Mirrors > Home > ILE Home > Th. List > dfif3 | Unicode version |
Description: Alternate definition of the conditional operator df-if 3470. 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 3471 | . 2 | |
2 | dfif3.1 | . . . . . 6 | |
3 | biidd 171 | . . . . . . 7 | |
4 | 3 | cbvabv 2262 | . . . . . 6 |
5 | 2, 4 | eqtri 2158 | . . . . 5 |
6 | 5 | ineq2i 3269 | . . . 4 |
7 | dfrab3 3347 | . . . 4 | |
8 | 6, 7 | eqtr4i 2161 | . . 3 |
9 | dfrab3 3347 | . . . 4 | |
10 | notab 3341 | . . . . . 6 | |
11 | 5 | difeq2i 3186 | . . . . . 6 |
12 | 10, 11 | eqtr4i 2161 | . . . . 5 |
13 | 12 | ineq2i 3269 | . . . 4 |
14 | 9, 13 | eqtr2i 2159 | . . 3 |
15 | 8, 14 | uneq12i 3223 | . 2 |
16 | 1, 15 | eqtr4i 2161 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wceq 1331 cab 2123 crab 2418 cvv 2681 cdif 3063 cun 3064 cin 3065 cif 3469 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-if 3470 |
This theorem is referenced by: (None) |
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