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Theorem hbs1 2183
Description:  x is not free in  [
y  /  x ] ph when  x and  y are distinct. (Contributed by NM, 5-Aug-1993.)
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 ax16 2051 . 2  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
2 hbsb2 2097 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
31, 2pm2.61i 159 1  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550   [wsb 1659
This theorem is referenced by:  nfs1v  2184  hbab1  2427  sb5ALT  28683  2sb5ndVD  29096  sb5ALTVD  29099  2sb5ndALT  29118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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