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Theorem dfpw12 4301
Description: Alternate expression for unit power classes. (Contributed by SF, 26-Jan-2015.)
Assertion
Ref Expression
dfpw12 1 SIk k k

Proof of Theorem dfpw12
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpw1 4144 . . 3 1
2 vex 2862 . . . . 5
32elimakv 4260 . . . 4 SIk k k SIk k
4 vex 2862 . . . . . . 7
5 opkelsikg 4264 . . . . . . 7 SIk k k
64, 2, 5mp2an 653 . . . . . 6 SIk k k
76exbii 1582 . . . . 5 SIk k k
8 exrot3 1744 . . . . 5 k k
97, 8bitr4i 243 . . . 4 SIk k k
10 df-3an 936 . . . . . . . . 9 k k
11 vex 2862 . . . . . . . . . . 11
12 vex 2862 . . . . . . . . . . 11
1311, 12opkelxpk 4248 . . . . . . . . . 10 k
1413anbi2i 675 . . . . . . . . 9 k
15 an4 797 . . . . . . . . 9
1610, 14, 153bitri 262 . . . . . . . 8 k
17162exbii 1583 . . . . . . 7 k
18 19.41vv 1902 . . . . . . 7
19 sneq 3744 . . . . . . . . . . . 12
20 eqeq12 2365 . . . . . . . . . . . 12
2119, 20sylan2 460 . . . . . . . . . . 11
22 eleq1 2413 . . . . . . . . . . . 12
2322adantl 452 . . . . . . . . . . 11
2421, 23anbi12d 691 . . . . . . . . . 10
252, 12, 24spc2ev 2947 . . . . . . . . 9
2625pm4.71ri 614 . . . . . . . 8
27 ancom 437 . . . . . . . 8
2826, 27bitr3i 242 . . . . . . 7
2917, 18, 283bitri 262 . . . . . 6 k
3029exbii 1582 . . . . 5 k
31 df-rex 2620 . . . . 5
3230, 31bitr4i 243 . . . 4 k
333, 9, 323bitri 262 . . 3 SIk k k
341, 33bitr4i 243 . 2 1 SIk k k
3534eqriv 2350 1 1 SIk k k
Colors of variables: wff setvar class
Syntax hints:   wb 176   wa 358   w3a 934  wex 1541   wceq 1642   wcel 1710  wrex 2615  cvv 2859  csn 3737  copk 4057  1 cpw1 4135   k cxpk 4174  kcimak 4179   SIk csik 4181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  ax-nin 4078  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-imak 4189  df-sik 4192
This theorem is referenced by:  pw1exg  4302
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