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Theorem dfsb7 2082
Description: An alternate definition of proper substitution df-sb 1884. By introducing a dummy variable  z in the definiens, we are able to eliminate any distinct variable restrictions among the variables  x,  y, and  ph of the definiendum. No distinct variable conflicts arise because  z effectively insulates  x from  y. To achieve this, we use a chain of two substitutions in the form of sb5 1994, first  z for  x then  y for  z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2243. Theorem sb7h 2084 provides a version where  ph and  z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
Assertion
Ref Expression
dfsb7  |-  ( [ y  /  x ] ph 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
Distinct variable groups:    x, z    y, z    ph, z
Allowed substitution hints:    ph( x, y)

Proof of Theorem dfsb7
StepHypRef Expression
1 sb5 1994 . . 3  |-  ( [ z  /  x ] ph 
<->  E. x ( x  =  z  /\  ph ) )
21sbbii 1886 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] E. x ( x  =  z  /\  ph ) )
3 nfv 1629 . . 3  |-  F/ z
ph
43sbco2 1981 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
5 sb5 1994 . 2  |-  ( [ y  /  z ] E. x ( x  =  z  /\  ph ) 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
62, 4, 53bitr3i 268 1  |-  ( [ y  /  x ] ph 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619   [wsb 1883
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884
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