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Theorem dvds1lem 12866
 Description: A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
dvds1lem.1
dvds1lem.2
dvds1lem.3
dvds1lem.4
Assertion
Ref Expression
dvds1lem
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem dvds1lem
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dvds1lem.3 . . . 4
2 dvds1lem.4 . . . 4
3 oveq1 6091 . . . . . 6
43eqeq1d 2446 . . . . 5
54rspcev 3054 . . . 4
61, 2, 5ee12an 1373 . . 3
76rexlimdva 2832 . 2
8 dvds1lem.1 . . 3
9 divides 12859 . . 3
108, 9syl 16 . 2
11 dvds1lem.2 . . 3
12 divides 12859 . . 3
1311, 12syl 16 . 2
147, 10, 133imtr4d 261 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  wrex 2708   class class class wbr 4215  (class class class)co 6084   cmul 9000  cz 10287   cdivides 12857 This theorem is referenced by:  negdvdsb  12871  dvdsnegb  12872  muldvds1  12879  muldvds2  12880  dvdscmul  12881  dvdsmulc  12882  dvdscmulr  12883  dvdsmulcr  12884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-iota 5421  df-fv 5465  df-ov 6087  df-dvds 12858
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