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Theorem sbiedh 1743
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1746). New proofs should use sbied 1744 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
sbiedh.1  |-  ( ph  ->  A. x ph )
sbiedh.2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
sbiedh.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
sbiedh  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)

Proof of Theorem sbiedh
StepHypRef Expression
1 sb1 1722 . . . 4  |-  ( [ y  /  x ] ps  ->  E. x ( x  =  y  /\  ps ) )
2 sbiedh.1 . . . . 5  |-  ( ph  ->  A. x ph )
3 sbiedh.3 . . . . . . 7  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
4 bi1 117 . . . . . . 7  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
53, 4syl6 33 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
65impd 252 . . . . 5  |-  ( ph  ->  ( ( x  =  y  /\  ps )  ->  ch ) )
72, 6eximdh 1573 . . . 4  |-  ( ph  ->  ( E. x ( x  =  y  /\  ps )  ->  E. x ch ) )
81, 7syl5 32 . . 3  |-  ( ph  ->  ( [ y  /  x ] ps  ->  E. x ch ) )
9 sbiedh.2 . . . 4  |-  ( ph  ->  ( ch  ->  A. x ch ) )
102, 919.9hd 1623 . . 3  |-  ( ph  ->  ( E. x ch 
->  ch ) )
118, 10syld 45 . 2  |-  ( ph  ->  ( [ y  /  x ] ps  ->  ch ) )
12 bi2 129 . . . . . . 7  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
133, 12syl6 33 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ( ch  ->  ps ) ) )
1413com23 78 . . . . 5  |-  ( ph  ->  ( ch  ->  (
x  =  y  ->  ps ) ) )
152, 14alimdh 1426 . . . 4  |-  ( ph  ->  ( A. x ch 
->  A. x ( x  =  y  ->  ps ) ) )
16 sb2 1723 . . . 4  |-  ( A. x ( x  =  y  ->  ps )  ->  [ y  /  x ] ps )
1715, 16syl6 33 . . 3  |-  ( ph  ->  ( A. x ch 
->  [ y  /  x ] ps ) )
189, 17syld 45 . 2  |-  ( ph  ->  ( ch  ->  [ y  /  x ] ps ) )
1911, 18impbid 128 1  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312   E.wex 1451   [wsb 1718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-sb 1719
This theorem is referenced by:  sbied  1744  sbieh  1746  sbcomxyyz  1921
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