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Theorem sb7f 1910
Description: This version of dfsb7 1909 does not require that  ph and  z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1460 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1687 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
sb7f.1  |-  ( ph  ->  A. z ph )
Assertion
Ref Expression
sb7f  |-  ( [ y  /  x ] ph 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sb7f
StepHypRef Expression
1 sb5 1809 . . 3  |-  ( [ z  /  x ] ph 
<->  E. x ( x  =  z  /\  ph ) )
21sbbii 1689 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] E. x ( x  =  z  /\  ph ) )
3 sb7f.1 . . 3  |-  ( ph  ->  A. z ph )
43sbco2v 1863 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
5 sb5 1809 . 2  |-  ( [ y  /  z ] E. x ( x  =  z  /\  ph ) 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
62, 4, 53bitr3i 208 1  |-  ( [ y  /  x ] ph 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   E.wex 1422   [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-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: (None)
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