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Theorem gcdcllem2 12967
 Description: Lemma for gcdn0cl 12969, gcddvds 12970 and dvdslegcd 12971. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
gcdcllem2.1
gcdcllem2.2
Assertion
Ref Expression
gcdcllem2
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem gcdcllem2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 breq1 4175 . . . . . 6
21ralbidv 2686 . . . . 5
3 gcdcllem2.1 . . . . 5
42, 3elrab2 3054 . . . 4
5 breq2 4176 . . . . . 6
6 breq2 4176 . . . . . 6
75, 6ralprg 3817 . . . . 5
87anbi2d 685 . . . 4
94, 8syl5bb 249 . . 3
10 breq1 4175 . . . . 5
11 breq1 4175 . . . . 5
1210, 11anbi12d 692 . . . 4
13 gcdcllem2.2 . . . 4
1412, 13elrab2 3054 . . 3
159, 14syl6rbbr 256 . 2
1615eqrdv 2402 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1649   wcel 1721  wral 2666  crab 2670  cpr 3775   class class class wbr 4172  cz 10238   cdivides 12807 This theorem is referenced by:  gcdcllem3  12968 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173
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