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Theorem isidlNEW 25549
 Description: The predicate "is an ideal of the ring ." (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
Hypotheses
Ref Expression
idlvalNEW.1
idlvalNEW.2
idlvalNEW.3
idlvalNEW.4
Assertion
Ref Expression
isidlNEW IdlNEW
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)   (,,)

Proof of Theorem isidlNEW
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 idlvalNEW.1 . . . 4
2 idlvalNEW.2 . . . 4
3 idlvalNEW.3 . . . 4
4 idlvalNEW.4 . . . 4
51, 2, 3, 4idlvalNEW 25548 . . 3 IdlNEW
65eleq2d 2363 . 2 IdlNEW
7 fvex 5555 . . . . . 6
83, 7eqeltri 2366 . . . . 5
98elpw2 4191 . . . 4
109anbi1i 676 . . 3
11 eleq2 2357 . . . . 5
12 eleq2 2357 . . . . . . . 8
1312raleqbi1dv 2757 . . . . . . 7
14 eleq2 2357 . . . . . . . . 9
15 eleq2 2357 . . . . . . . . 9
1614, 15anbi12d 691 . . . . . . . 8
1716ralbidv 2576 . . . . . . 7
1813, 17anbi12d 691 . . . . . 6
1918raleqbi1dv 2757 . . . . 5
2011, 19anbi12d 691 . . . 4
2120elrab 2936 . . 3
22 3anass 938 . . 3
2310, 21, 223bitr4i 268 . 2
246, 23syl6bb 252 1 IdlNEW
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3a 934   wceq 1632   wcel 1696  wral 2556  crab 2560  cvv 2801   wss 3165  cpw 3638  cfv 5271  (class class class)co 5874  cbs 13164   cplusg 13224  cmulr 13225  c0g 13416  crg 15353  IdlNEWcidln 25546 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-idlNEW 25547
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