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Theorem sb7af 1911
Description: An alternate definition of proper substitution df-sb 1687. Similar to dfsb7a 1912 but does not require that  ph and  z be distinct. Similar to sb7f 1910 in that it involves a dummy variable  z, but expressed in terms of  A. rather than  E.. (Contributed by Jim Kingdon, 5-Feb-2018.)
Hypothesis
Ref Expression
sb7af.1  |-  F/ z
ph
Assertion
Ref Expression
sb7af  |-  ( [ y  /  x ] ph 
<-> 
A. z ( z  =  y  ->  A. x
( x  =  z  ->  ph ) ) )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sb7af
StepHypRef Expression
1 sb6 1808 . . 3  |-  ( [ z  /  x ] ph 
<-> 
A. x ( x  =  z  ->  ph )
)
21sbbii 1689 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] A. x ( x  =  z  ->  ph )
)
3 sb7af.1 . . 3  |-  F/ z
ph
43sbco2 1881 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
5 sb6 1808 . 2  |-  ( [ y  /  z ] A. x ( x  =  z  ->  ph )  <->  A. z ( z  =  y  ->  A. x
( x  =  z  ->  ph ) ) )
62, 4, 53bitr3i 208 1  |-  ( [ y  /  x ] ph 
<-> 
A. z ( z  =  y  ->  A. x
( x  =  z  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   F/wnf 1390   [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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  dfsb7a  1912
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