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Theorem sbh 1706
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)
Hypothesis
Ref Expression
sbh.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
sbh  |-  ( [ y  /  x ] ph 
<-> 
ph )

Proof of Theorem sbh
StepHypRef Expression
1 sb1 1696 . . . 4  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 sbh.1 . . . . 5  |-  ( ph  ->  A. x ph )
3219.41h 1620 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  <->  ( E. x  x  =  y  /\  ph )
)
41, 3sylib 120 . . 3  |-  ( [ y  /  x ] ph  ->  ( E. x  x  =  y  /\  ph ) )
54simprd 112 . 2  |-  ( [ y  /  x ] ph  ->  ph )
6 stdpc4 1705 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
72, 6syl 14 . 2  |-  ( ph  ->  [ y  /  x ] ph )
85, 7impbii 124 1  |-  ( [ y  /  x ] ph 
<-> 
ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   E.wex 1426   [wsb 1692
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-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by:  sbf  1707  sb6x  1709  nfs1f  1710  hbs1f  1711  sbid2h  1777  sblimv  1822  sbrim  1878  sbrbif  1884  elsb3  1900  elsb4  1901
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