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Theorem dfsb7 2205
Description: An alternate definition of proper substitution df-sb 1661. 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 2183, 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 2430. Theorem sb7h 2204 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 1631 . 2  |-  F/ z
ph
21sb7f 2203 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 1551   [wsb 1660
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 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661
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