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Theorem dvds2lem 14008
 Description: A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
dvds2lem.1
dvds2lem.2
dvds2lem.3
dvds2lem.4
dvds2lem.5
Assertion
Ref Expression
dvds2lem
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem dvds2lem
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dvds2lem.1 . . . . . 6
2 dvds2lem.2 . . . . . 6
3 divides 14000 . . . . . . 7
4 divides 14000 . . . . . . 7
53, 4bi2anan9 873 . . . . . 6
61, 2, 5syl2anc 661 . . . . 5
76biimpd 207 . . . 4
8 reeanv 3025 . . . 4
97, 8syl6ibr 227 . . 3
10 dvds2lem.4 . . . . 5
11 dvds2lem.5 . . . . 5
12 oveq1 6303 . . . . . . 7
1312eqeq1d 2459 . . . . . 6
1413rspcev 3210 . . . . 5
1510, 11, 14syl6an 545 . . . 4
1615rexlimdvva 2956 . . 3
179, 16syld 44 . 2
18 dvds2lem.3 . . 3
19 divides 14000 . . 3
2018, 19syl 16 . 2
2117, 20sylibrd 234 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  wrex 2808   class class class wbr 4456  (class class class)co 6296   cmul 9514  cz 10885   cdvds 13998 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  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 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-iota 5557  df-fv 5602  df-ov 6299  df-dvds 13999 This theorem is referenced by:  dvds2ln  14026  dvds2add  14027  dvds2sub  14028  dvdstr  14030
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