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Mirrors > Home > NFE Home > Th. List > phiun | Unicode version |
Description: The phi operation distributes over union. (Contributed by SF, 20-Feb-2015.) |
Ref | Expression |
---|---|
phiun | Phi Phi Phi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexun 3443 | . . 3 Nn 1c Nn 1c Nn 1c | |
2 | 1 | abbii 2465 | . 2 Nn 1c Nn 1c Nn 1c |
3 | df-phi 4565 | . 2 Phi Nn 1c | |
4 | df-phi 4565 | . . . 4 Phi Nn 1c | |
5 | df-phi 4565 | . . . 4 Phi Nn 1c | |
6 | 4, 5 | uneq12i 3416 | . . 3 Phi Phi Nn 1c Nn 1c |
7 | unab 3521 | . . 3 Nn 1c Nn 1c Nn 1c Nn 1c | |
8 | 6, 7 | eqtri 2373 | . 2 Phi Phi Nn 1c Nn 1c |
9 | 2, 3, 8 | 3eqtr4i 2383 | 1 Phi Phi Phi |
Colors of variables: wff setvar class |
Syntax hints: wo 357 wceq 1642 wcel 1710 cab 2339 wrex 2615 cun 3207 cif 3662 1cc1c 4134 Nn cnnc 4373 cplc 4375 Phi cphi 4562 |
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-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-phi 4565 |
This theorem is referenced by: phialllem2 4617 |
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