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Theorem dfsb7 2058
Description: An alternate definition of proper substitution df-sb 1630. 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 2039, 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 2270. Theorem sb7h 2060 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 2039 . . 3  |-  ( [ z  /  x ] ph 
<->  E. x ( x  =  z  /\  ph ) )
21sbbii 1634 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] E. x ( x  =  z  /\  ph ) )
3 nfv 1605 . . 3  |-  F/ z
ph
43sbco2 2026 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
5 sb5 2039 . 2  |-  ( [ y  /  z ] E. x ( x  =  z  /\  ph ) 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
62, 4, 53bitr3i 266 1  |-  ( [ y  /  x ] ph 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528   [wsb 1629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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