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Theorem dvds2lem 12862
 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 12854 . . . . . . 7
4 divides 12854 . . . . . . 7
53, 4bi2anan9 844 . . . . . 6
61, 2, 5syl2anc 643 . . . . 5
76biimpd 199 . . . 4
8 reeanv 2875 . . . 4
97, 8syl6ibr 219 . . 3
10 dvds2lem.4 . . . . 5
11 dvds2lem.5 . . . . 5
12 oveq1 6088 . . . . . . 7
1312eqeq1d 2444 . . . . . 6
1413rspcev 3052 . . . . 5
1510, 11, 14ee12an 1372 . . . 4
1615rexlimdvva 2837 . . 3
179, 16syld 42 . 2
18 dvds2lem.3 . . 3
19 divides 12854 . . 3
2018, 19syl 16 . 2
2117, 20sylibrd 226 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wrex 2706   class class class wbr 4212  (class class class)co 6081   cmul 8995  cz 10282   cdivides 12852 This theorem is referenced by:  dvds2ln  12880  dvds2add  12881  dvds2sub  12882  dvdstr  12883 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-iota 5418  df-fv 5462  df-ov 6084  df-dvds 12853
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