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Theorem keephyp2v 3796
Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3782). (Contributed by NM, 16-Apr-2005.)
Hypotheses
Ref Expression
keephyp2v.1  |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch )
)
keephyp2v.2  |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch  <->  th )
)
keephyp2v.3  |-  ( C  =  if ( ph ,  A ,  C )  ->  ( ta  <->  et )
)
keephyp2v.4  |-  ( D  =  if ( ph ,  B ,  D )  ->  ( et  <->  th )
)
keephyp2v.5  |-  ps
keephyp2v.6  |-  ta
Assertion
Ref Expression
keephyp2v  |-  th

Proof of Theorem keephyp2v
StepHypRef Expression
1 keephyp2v.5 . . 3  |-  ps
2 iftrue 3747 . . . . . 6  |-  ( ph  ->  if ( ph ,  A ,  C )  =  A )
32eqcomd 2443 . . . . 5  |-  ( ph  ->  A  =  if (
ph ,  A ,  C ) )
4 keephyp2v.1 . . . . 5  |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch )
)
53, 4syl 16 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
6 iftrue 3747 . . . . . 6  |-  ( ph  ->  if ( ph ,  B ,  D )  =  B )
76eqcomd 2443 . . . . 5  |-  ( ph  ->  B  =  if (
ph ,  B ,  D ) )
8 keephyp2v.2 . . . . 5  |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch  <->  th )
)
97, 8syl 16 . . . 4  |-  ( ph  ->  ( ch  <->  th )
)
105, 9bitrd 246 . . 3  |-  ( ph  ->  ( ps  <->  th )
)
111, 10mpbii 204 . 2  |-  ( ph  ->  th )
12 keephyp2v.6 . . 3  |-  ta
13 iffalse 3748 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  A ,  C )  =  C )
1413eqcomd 2443 . . . . 5  |-  ( -. 
ph  ->  C  =  if ( ph ,  A ,  C ) )
15 keephyp2v.3 . . . . 5  |-  ( C  =  if ( ph ,  A ,  C )  ->  ( ta  <->  et )
)
1614, 15syl 16 . . . 4  |-  ( -. 
ph  ->  ( ta  <->  et )
)
17 iffalse 3748 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  B ,  D )  =  D )
1817eqcomd 2443 . . . . 5  |-  ( -. 
ph  ->  D  =  if ( ph ,  B ,  D ) )
19 keephyp2v.4 . . . . 5  |-  ( D  =  if ( ph ,  B ,  D )  ->  ( et  <->  th )
)
2018, 19syl 16 . . . 4  |-  ( -. 
ph  ->  ( et  <->  th )
)
2116, 20bitrd 246 . . 3  |-  ( -. 
ph  ->  ( ta  <->  th )
)
2212, 21mpbii 204 . 2  |-  ( -. 
ph  ->  th )
2311, 22pm2.61i 159 1  |-  th
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    = wceq 1653   ifcif 3741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-if 3742
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