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Theorem sb7f 2072
Description: This version of dfsb7 2071 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-17 1606 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1639 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb7f.1  |-  F/ z
ph
Assertion
Ref Expression
sb7f  |-  ( [ y  /  x ] ph 
<->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
) )
Distinct variable groups:    x, z    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sb7f
StepHypRef Expression
1 sb5 2052 . . 3  |-  ( [ z  /  x ] ph 
<->  E. x ( x  =  z  /\  ph ) )
21sbbii 1643 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] E. x ( x  =  z  /\  ph ) )
3 sb7f.1 . . 3  |-  F/ z
ph
43sbco2 2039 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
5 sb5 2052 . 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 1531   F/wnf 1534   [wsb 1638
This theorem is referenced by:  sb7h  2073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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