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Theorem sb7h 2303
Description: This version of dfsb7 2304 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-5 1766 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1806 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
sb7h.1  |-  ( ph  ->  A. z ph )
Assertion
Ref Expression
sb7h  |-  ( [ y  /  x ] ph 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sb7h
StepHypRef Expression
1 sb7h.1 . . 3  |-  ( ph  ->  A. z ph )
21nfi 1682 . 2  |-  F/ z
ph
32sb7f 2302 1  |-  ( [ y  /  x ] ph 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450   E.wex 1671   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by: (None)
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