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Theorem dvds1lem 13995
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:   ,J   ,   ,M   ,N   ,

Proof of Theorem dvds1lem
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dvds1lem.3 . . . 4
2 dvds1lem.4 . . . 4
3 oveq1 6303 . . . . . 6
43eqeq1d 2459 . . . . 5
54rspcev 3210 . . . 4
61, 2, 5syl6an 545 . . 3
76rexlimdva 2949 . 2
8 dvds1lem.1 . . 3
9 divides 13988 . . 3
108, 9syl 16 . 2
11 dvds1lem.2 . . 3
12 divides 13988 . . 3
1311, 12syl 16 . 2
147, 10, 133imtr4d 268 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  E.wrex 2808   class class class wbr 4452  (class class class)co 6296   cmul 9518   cz 10889   cdvds 13986
This theorem is referenced by:  negdvdsb  14000  dvdsnegb  14001  muldvds1  14008  muldvds2  14009  dvdscmul  14010  dvdsmulc  14011  dvdscmulr  14012  dvdsmulcr  14013
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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
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-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  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-uni 4250  df-br 4453  df-opab 4511  df-iota 5556  df-fv 5601  df-ov 6299  df-dvds 13987
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