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Theorem dfsb7 2179
Description: An alternate definition of proper substitution df-sb 1656. 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 2157, 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 2399. Theorem sb7h 2178 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:    y, z    ph, z
Allowed substitution hints:    ph( x, y)

Proof of Theorem dfsb7
StepHypRef Expression
1 nfv 1626 . 2  |-  F/ z
ph
21sb7f 2177 1  |-  ( [ y  /  x ] ph 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547   [wsb 1655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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