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Mirrors > Home > NFE Home > Th. List > dfif3 | Unicode version |
Description: Alternate definition of the conditional operator df-if 3663. 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 3665 | . 2 | |
2 | dfif3.1 | . . . . . 6 | |
3 | biidd 228 | . . . . . . 7 | |
4 | 3 | cbvabv 2472 | . . . . . 6 |
5 | 2, 4 | eqtri 2373 | . . . . 5 |
6 | 5 | ineq2i 3454 | . . . 4 |
7 | dfrab3 3531 | . . . 4 | |
8 | 6, 7 | eqtr4i 2376 | . . 3 |
9 | dfrab3 3531 | . . . 4 | |
10 | notab 3525 | . . . . . 6 | |
11 | 5 | difeq2i 3382 | . . . . . 6 |
12 | 10, 11 | eqtr4i 2376 | . . . . 5 |
13 | 12 | ineq2i 3454 | . . . 4 |
14 | 9, 13 | eqtr2i 2374 | . . 3 |
15 | 8, 14 | uneq12i 3416 | . 2 |
16 | 1, 15 | eqtr4i 2376 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wceq 1642 cab 2339 crab 2618 cvv 2859 cdif 3206 cun 3207 cin 3208 cif 3662 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-if 3663 |
This theorem is referenced by: dfif4 3673 |
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