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Theorem wl-lem-moexsb 31604
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 1954 . . 3  |-  F/ x A. x ( ph  ->  x  =  z )
2 nfs1v 2233 . . 3  |-  F/ x [ z  /  x ] ph
3 sp 1912 . . . . 5  |-  ( A. x ( ph  ->  x  =  z )  -> 
( ph  ->  x  =  z ) )
4 ax12v 1908 . . . . 5  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
53, 4syli 38 . . . 4  |-  ( A. x ( ph  ->  x  =  z )  -> 
( ph  ->  A. x
( x  =  z  ->  ph ) ) )
6 sb2 2147 . . . 4  |-  ( A. x ( x  =  z  ->  ph )  ->  [ z  /  x ] ph )
75, 6syl6 34 . . 3  |-  ( A. x ( ph  ->  x  =  z )  -> 
( ph  ->  [ z  /  x ] ph ) )
81, 2, 7exlimd 1972 . 2  |-  ( A. x ( ph  ->  x  =  z )  -> 
( E. x ph  ->  [ z  /  x ] ph ) )
9 spsbe 1793 . 2  |-  ( [ z  /  x ] ph  ->  E. x ph )
108, 9impbid1 206 1  |-  ( A. x ( ph  ->  x  =  z )  -> 
( E. x ph  <->  [ z  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435   E.wex 1659   [wsb 1789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1790
This theorem is referenced by: (None)
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