Theorem keephyp 4152

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

Hypotheses

Ref	Expression
keephyp.1	|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )
keephyp.2	|- ( B = if ( ph , A , B ) -> ( ch <-> th ) )
keephyp.3	|- ps
keephyp.4	|- ch

Assertion

Ref	Expression
keephyp	|- th

Proof of Theorem keephyp

Step	Hyp	Ref	Expression
1		keephyp.3	. 2 |- ps
2		keephyp.4	. 2 |- ch
3		keephyp.1	. . 3 |- ( A = if ( ph , A , B ) -> ( ps <-> th ) )
4		keephyp.2	. . 3 |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )
5	3, 4	ifboth 4124	. 2 |- ( ( ps /\ ch ) -> th )
6	1, 2, 5	mp2an 708	1 |- th

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  keepel  4155  boxcutc  7951  fin23lem13  9154  abvtrivd  18840  znf1o  19900  zntoslem  19905  dscmet  22377  sqff1o  24908  lgsne0  25060  dchrisum0flblem1  25197  dchrisum0flblem2  25198