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Theorem phidisjnn 4616
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 3592 . . . . . . . . . 10 Nn Nn
21biimpi 186 . . . . . . . . 9 Nn Nn
32r19.21bi 2713 . . . . . . . 8 Nn Nn
4 iffalse 3670 . . . . . . . 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 2632 . . . 4 Nn Nn 1c
10 risset 2662 . . . 4
119, 10syl6bbr 254 . . 3 Nn Nn 1c
1211alrimiv 1631 . 2 Nn Nn 1c
13 df-phi 4566 . . . 4 Phi Nn 1c
1413eqeq1i 2360 . . 3 Phi Nn 1c
15 abeq1 2460 . . 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 2615  wrex 2616   cin 3209  c0 3551  cif 3663  1cc1c 4135   Nn cnnc 4374   cplc 4376   Phi cphi 4563
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
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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-if 3664  df-phi 4566
This theorem is referenced by:  phialllem2  4618
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