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Theorem hbsbd 1903
Description: Deduction version of hbsb 1868. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
Hypotheses
Ref Expression
hbsbd.1  |-  ( ph  ->  A. x ph )
hbsbd.2  |-  ( ph  ->  A. z ph )
hbsbd.3  |-  ( ph  ->  ( ps  ->  A. z ps ) )
Assertion
Ref Expression
hbsbd  |-  ( ph  ->  ( [ y  /  x ] ps  ->  A. z [ y  /  x ] ps ) )
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem hbsbd
StepHypRef Expression
1 hbsbd.2 . . . 4  |-  ( ph  ->  A. z ph )
21nfi 1394 . . 3  |-  F/ z
ph
3 hbsbd.3 . . . . . . 7  |-  ( ph  ->  ( ps  ->  A. z ps ) )
41, 3nfdh 1460 . . . . . 6  |-  ( ph  ->  F/ z ps )
52, 4nfim1 1506 . . . . 5  |-  F/ z ( ph  ->  ps )
65nfsb 1867 . . . 4  |-  F/ z [ y  /  x ] ( ph  ->  ps )
7 hbsbd.1 . . . . . 6  |-  ( ph  ->  A. x ph )
87sbrim 1875 . . . . 5  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
98nfbii 1405 . . . 4  |-  ( F/ z [ y  /  x ] ( ph  ->  ps )  <->  F/ z ( ph  ->  [ y  /  x ] ps ) )
106, 9mpbi 143 . . 3  |-  F/ z ( ph  ->  [ y  /  x ] ps )
112, 10nfrimi 1461 . 2  |-  ( ph  ->  F/ z [ y  /  x ] ps )
1211nfrd 1456 1  |-  ( ph  ->  ( [ y  /  x ] ps  ->  A. z [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1285   F/wnf 1392   [wsb 1689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690
This theorem is referenced by: (None)
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