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Theorem wl-lem-exsb 31965
Description: 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-exsb  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem wl-lem-exsb
StepHypRef Expression
1 ax12v2 1952 . 2  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
2 sp 1957 . . 3  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
32com12 31 . 2  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
41, 3impbid 195 1  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450
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-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672
This theorem is referenced by: (None)
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