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Theorem hbs1 1856
Description:  x is not free in  [
y  /  x ] ph when  x and  y are distinct. (Contributed by NM, 5-Aug-1993.) (Proof by Jim Kingdon, 16-Dec-2017.) (New usage is discouraged.)
Assertion
Ref Expression
hbs1  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem hbs1
StepHypRef Expression
1 sb6 1808 . 2  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
2 ax-ial 1468 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x A. x ( x  =  y  ->  ph ) )
31, 2hbxfrbi 1402 1  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283   [wsb 1686
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  nfs1v  1857  sb9v  1896  eu1  1967  mopick  2020  hbab1  2071
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