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Theorem gcdcllem2 14150
Description: Lemma for gcdn0cl 14152, gcddvds 14153 and dvdslegcd 14154. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
gcdcllem2.1
gcdcllem2.2
Assertion
Ref Expression
gcdcllem2
Distinct variable groups:   , ,M   ,N,

Proof of Theorem gcdcllem2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 breq1 4455 . . . . . 6
21ralbidv 2896 . . . . 5
3 gcdcllem2.1 . . . . 5
42, 3elrab2 3259 . . . 4
5 breq2 4456 . . . . . 6
6 breq2 4456 . . . . . 6
75, 6ralprg 4078 . . . . 5
87anbi2d 703 . . . 4
94, 8syl5bb 257 . . 3
10 breq1 4455 . . . . 5
11 breq1 4455 . . . . 5
1210, 11anbi12d 710 . . . 4
13 gcdcllem2.2 . . . 4
1412, 13elrab2 3259 . . 3
159, 14syl6rbbr 264 . 2
1615eqrdv 2454 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  {crab 2811  {cpr 4031   class class class wbr 4452   cz 10889   cdvds 13986
This theorem is referenced by:  gcdcllem3  14151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453
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