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Theorem wl-lem-moexsb 31967
Description: The antecedent  A. x
( ph  ->  x  =  z ) relates to  E* x ph, but is better suited for usage in proofs. Note that no distinct variable restriction is placed on 
ph.

This theorem provides a basic working step in proving theorems about  E* or  E!. (Contributed by Wolf Lammen, 3-Oct-2019.)

Assertion
Ref Expression
wl-lem-moexsb  |-  ( A. x ( ph  ->  x  =  z )  -> 
( E. x ph  <->  [ z  /  x ] ph ) )
Distinct variable group:    x, z
Allowed substitution hints:    ph( x, z)

Proof of Theorem wl-lem-moexsb
StepHypRef Expression
1 nfa1 1999 . . 3  |-  F/ x A. x ( ph  ->  x  =  z )
2 nfs1v 2286 . . 3  |-  F/ x [ z  /  x ] ph
3 sp 1957 . . . . 5  |-  ( A. x ( ph  ->  x  =  z )  -> 
( ph  ->  x  =  z ) )
4 ax12v2 1952 . . . . 5  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
53, 4syli 37 . . . 4  |-  ( A. x ( ph  ->  x  =  z )  -> 
( ph  ->  A. x
( x  =  z  ->  ph ) ) )
6 sb2 2201 . . . 4  |-  ( A. x ( x  =  z  ->  ph )  ->  [ z  /  x ] ph )
75, 6syl6 33 . . 3  |-  ( A. x ( ph  ->  x  =  z )  -> 
( ph  ->  [ z  /  x ] ph ) )
81, 2, 7exlimd 2017 . 2  |-  ( A. x ( ph  ->  x  =  z )  -> 
( E. x ph  ->  [ z  /  x ] ph ) )
9 spsbe 1809 . 2  |-  ( [ z  /  x ] ph  ->  E. x ph )
108, 9impbid1 208 1  |-  ( A. x ( ph  ->  x  =  z )  -> 
( E. x ph  <->  [ z  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   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-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by: (None)
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