Theorem elimhyp 4000

Description: Eliminate a hypothesis containing class variable when it is known for a specific class . For more information, see comments in dedth 3993. (Contributed by NM, 15-May-1999.)

Hypotheses

Ref	Expression
elimhyp.1
elimhyp.2
elimhyp.3

Assertion

Ref	Expression
elimhyp

Proof of Theorem elimhyp

Step	Hyp	Ref	Expression
1		iftrue 3947	. . . . 5
2	1	eqcomd 2465	. . . 4
3		elimhyp.1	. . . 4
4	2, 3	syl 16	. . 3
5	4	ibi 241	. 2
6		elimhyp.3	. . 3
7		iffalse 3950	. . . . 5
8	7	eqcomd 2465	. . . 4
9		elimhyp.2	. . . 4
10	8, 9	syl 16	. . 3
11	6, 10	mpbii 211	. 2
12	5, 11	pm2.61i 164	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  =wceq 1395  ifcif 3941

This theorem is referenced by:  elimel  4004  elimf  5735  oeoa  7265  oeoe  7267  limensuc  7714  axcc4dom  8842  elimne0  9607  elimgt0  10403  elimge0  10404  2ndcdisj  19957  siilem2  25767  normlem7tALT  26036  hhsssh  26185  shintcl  26248  chintcl  26250  spanun  26463  elunop2  26932  lnophm  26938  nmbdfnlb  26969  hmopidmch  27072  hmopidmpj  27073  chirred  27314  limsucncmp  29911  elimhyps  34692  elimhyps2  34695

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-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-if 3942