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Theorem keephyp2v 3622
Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3608). (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 3573 . . . . . 6  |-  ( ph  ->  if ( ph ,  A ,  C )  =  A )
32eqcomd 2290 . . . . 5  |-  ( ph  ->  A  =  if (
ph ,  A ,  C ) )
4 keephyp2v.1 . . . . 5  |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch )
)
53, 4syl 15 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
6 iftrue 3573 . . . . . 6  |-  ( ph  ->  if ( ph ,  B ,  D )  =  B )
76eqcomd 2290 . . . . 5  |-  ( ph  ->  B  =  if (
ph ,  B ,  D ) )
8 keephyp2v.2 . . . . 5  |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch  <->  th )
)
97, 8syl 15 . . . 4  |-  ( ph  ->  ( ch  <->  th )
)
105, 9bitrd 244 . . 3  |-  ( ph  ->  ( ps  <->  th )
)
111, 10mpbii 202 . 2  |-  ( ph  ->  th )
12 keephyp2v.6 . . 3  |-  ta
13 iffalse 3574 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  A ,  C )  =  C )
1413eqcomd 2290 . . . . 5  |-  ( -. 
ph  ->  C  =  if ( ph ,  A ,  C ) )
15 keephyp2v.3 . . . . 5  |-  ( C  =  if ( ph ,  A ,  C )  ->  ( ta  <->  et )
)
1614, 15syl 15 . . . 4  |-  ( -. 
ph  ->  ( ta  <->  et )
)
17 iffalse 3574 . . . . . 6  |-  ( -. 
ph  ->  if ( ph ,  B ,  D )  =  D )
1817eqcomd 2290 . . . . 5  |-  ( -. 
ph  ->  D  =  if ( ph ,  B ,  D ) )
19 keephyp2v.4 . . . . 5  |-  ( D  =  if ( ph ,  B ,  D )  ->  ( et  <->  th )
)
2018, 19syl 15 . . . 4  |-  ( -. 
ph  ->  ( et  <->  th )
)
2116, 20bitrd 244 . . 3  |-  ( -. 
ph  ->  ( ta  <->  th )
)
2212, 21mpbii 202 . 2  |-  ( -. 
ph  ->  th )
2311, 22pm2.61i 156 1  |-  th
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1625   ifcif 3567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-if 3568
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