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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 ((ANn ) = Phi A = A)

Proof of Theorem phidisjnn
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disj 3591 . . . . . . . . . 10 ((ANn ) = y A ¬ y Nn )
21biimpi 186 . . . . . . . . 9 ((ANn ) = y A ¬ y Nn )
32r19.21bi 2712 . . . . . . . 8 (((ANn ) = y A) → ¬ y Nn )
4 iffalse 3669 . . . . . . . 8 y Nn → if(y Nn , (y +c 1c), y) = y)
53, 4syl 15 . . . . . . 7 (((ANn ) = y A) → if(y Nn , (y +c 1c), y) = y)
65eqeq2d 2364 . . . . . 6 (((ANn ) = y A) → (x = if(y Nn , (y +c 1c), y) ↔ x = y))
7 equcom 1680 . . . . . 6 (y = xx = y)
86, 7syl6bbr 254 . . . . 5 (((ANn ) = y A) → (x = if(y Nn , (y +c 1c), y) ↔ y = x))
98rexbidva 2631 . . . 4 ((ANn ) = → (y A x = if(y Nn , (y +c 1c), y) ↔ y A y = x))
10 risset 2661 . . . 4 (x Ay A y = x)
119, 10syl6bbr 254 . . 3 ((ANn ) = → (y A x = if(y Nn , (y +c 1c), y) ↔ x A))
1211alrimiv 1631 . 2 ((ANn ) = x(y A x = if(y Nn , (y +c 1c), y) ↔ x A))
13 df-phi 4565 . . . 4 Phi A = {x y A x = if(y Nn , (y +c 1c), y)}
1413eqeq1i 2360 . . 3 ( Phi A = A ↔ {x y A x = if(y Nn , (y +c 1c), y)} = A)
15 abeq1 2459 . . 3 ({x y A x = if(y Nn , (y +c 1c), y)} = Ax(y A x = if(y Nn , (y +c 1c), y) ↔ x A))
1614, 15bitri 240 . 2 ( Phi A = Ax(y A x = if(y Nn , (y +c 1c), y) ↔ x A))
1712, 16sylibr 203 1 ((ANn ) = Phi A = A)
 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   ifcif 3662  1cc1c 4134   Nn cnnc 4373   +c 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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