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Theorem pm5.21ndd 654
Description: Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.)
Hypotheses
Ref Expression
pm5.21ndd.1  |-  ( ph  ->  ( ch  ->  ps ) )
pm5.21ndd.2  |-  ( ph  ->  ( th  ->  ps ) )
pm5.21ndd.3  |-  ( ph  ->  ( ps  ->  ( ch 
<->  th ) ) )
Assertion
Ref Expression
pm5.21ndd  |-  ( ph  ->  ( ch  <->  th )
)

Proof of Theorem pm5.21ndd
StepHypRef Expression
1 pm5.21ndd.1 . . . 4  |-  ( ph  ->  ( ch  ->  ps ) )
2 pm5.21ndd.3 . . . 4  |-  ( ph  ->  ( ps  ->  ( ch 
<->  th ) ) )
31, 2syld 44 . . 3  |-  ( ph  ->  ( ch  ->  ( ch 
<->  th ) ) )
43ibd 176 . 2  |-  ( ph  ->  ( ch  ->  th )
)
5 pm5.21ndd.2 . . . . 5  |-  ( ph  ->  ( th  ->  ps ) )
65, 2syld 44 . . . 4  |-  ( ph  ->  ( th  ->  ( ch 
<->  th ) ) )
7 bicom1 129 . . . 4  |-  ( ( ch  <->  th )  ->  ( th 
<->  ch ) )
86, 7syl6 33 . . 3  |-  ( ph  ->  ( th  ->  ( th 
<->  ch ) ) )
98ibd 176 . 2  |-  ( ph  ->  ( th  ->  ch ) )
104, 9impbid 127 1  |-  ( ph  ->  ( ch  <->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  pm5.21nd  859  sbcrext  2892  rmob  2907  epelg  4053  eqbrrdva  4533  relbrcnvg  4734  fmptco  5362  ovelrn  5680  brtpos2  5900  brdomg  6295  genpelvl  6764  genpelvu  6765  fzoval  9235  clim  10258
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