Theorem gcdcllem2 14150

Description: Lemma for gcdn0cl 14152, gcddvds 14153 and dvdslegcd 14154. (Contributed by Paul Chapman, 21-Mar-2011.)

Hypotheses

Ref	Expression
gcdcllem2.1
gcdcllem2.2

Assertion

Ref	Expression
gcdcllem2

Distinct variable groups:   ,,M   ,N,

Proof of Theorem gcdcllem2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4455	. . . . . 6
2	1	ralbidv 2896	. . . . 5
3		gcdcllem2.1	. . . . 5
4	2, 3	elrab2 3259	. . . 4
5		breq2 4456	. . . . . 6
6		breq2 4456	. . . . . 6
7	5, 6	ralprg 4078	. . . . 5
8	7	anbi2d 703	. . . 4
9	4, 8	syl5bb 257	. . 3
10		breq1 4455	. . . . 5
11		breq1 4455	. . . . 5
12	10, 11	anbi12d 710	. . . 4
13		gcdcllem2.2	. . . 4
14	12, 13	elrab2 3259	. . 3
15	9, 14	syl6rbbr 264	. 2
16	15	eqrdv 2454	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  {crab 2811  {cpr 4031   class class class wbr 4452  cz 10889  cdvds 13986

This theorem is referenced by:  gcdcllem3  14151

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-sbc 3328  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-br 4453