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Theorem phidisjnn 4615
 Description: The phi operation applied to a set disjoint from the naturals has no effect. (Contributed by SF, 20-Feb-2015.)
Assertion
Ref Expression
phidisjnn Nn Phi

Proof of Theorem phidisjnn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disj 3591 . . . . . . . . . 10 Nn Nn
21biimpi 186 . . . . . . . . 9 Nn Nn
32r19.21bi 2712 . . . . . . . 8 Nn Nn
4 iffalse 3669 . . . . . . . 8 Nn Nn 1c
53, 4syl 15 . . . . . . 7 Nn Nn 1c
65eqeq2d 2364 . . . . . 6 Nn Nn 1c
7 equcom 1680 . . . . . 6
86, 7syl6bbr 254 . . . . 5 Nn Nn 1c
98rexbidva 2631 . . . 4 Nn Nn 1c
10 risset 2661 . . . 4
119, 10syl6bbr 254 . . 3 Nn Nn 1c
1211alrimiv 1631 . 2 Nn Nn 1c
13 df-phi 4565 . . . 4 Phi Nn 1c
1413eqeq1i 2360 . . 3 Phi Nn 1c
15 abeq1 2459 . . 3 Nn 1c Nn 1c
1614, 15bitri 240 . 2 Phi Nn 1c
1712, 16sylibr 203 1 Nn Phi
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1540   wceq 1642   wcel 1710  cab 2339  wral 2614  wrex 2615   cin 3208  c0 3550  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-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-if 3663  df-phi 4565 This theorem is referenced by:  phialllem2  4617
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