Theorem keephyp 4006

Description: Transform a hypothesis that we want to keep (but contains the same class variable used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)

Hypotheses

Ref	Expression
keephyp.1
keephyp.2
keephyp.3
keephyp.4

Assertion

Ref	Expression
keephyp

Proof of Theorem keephyp

Step	Hyp	Ref	Expression
1		keephyp.3	. 2
2		keephyp.4	. 2
3		keephyp.1	. . 3
4		keephyp.2	. . 3
5	3, 4	ifboth 3977	. 2
6	1, 2, 5	mp2an 672	1

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

This theorem is referenced by:  keepel  4009  boxcutc  7532  fin23lem13  8733  abvtrivd  17489  znf1o  18590  zntoslem  18595  dscmet  21093  sqff1o  23456  lgsne0  23608  dchrisum0flblem1  23693  dchrisum0flblem2  23694

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