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Theorem sb5ALT 28587
Description: Equivalence for substitution. Alternate proof of sb5 2052. This proof is sb5ALTVD 29005 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sb5ALT  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb5ALT
StepHypRef Expression
1 equsb1 1987 . . . 4  |-  [ y  /  x ] x  =  y
2 sban 2022 . . . . 5  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  <->  ( [
y  /  x ]
x  =  y  /\  [ y  /  x ] ph ) )
32simplbi2com 1364 . . . 4  |-  ( [ y  /  x ] ph  ->  ( [ y  /  x ] x  =  y  ->  [ y  /  x ] ( x  =  y  /\  ph ) ) )
41, 3mpi 16 . . 3  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ( x  =  y  /\  ph )
)
5 spsbe 2028 . . 3  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  ph )
)
64, 5syl 15 . 2  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
7 hbs1 2057 . . 3  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
8 simpr 447 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  ph )
98a1i 10 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  ( ( x  =  y  /\  ph )  ->  ph ) )
10 simpl 443 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  x  =  y )
1110a1i 10 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  ( ( x  =  y  /\  ph )  ->  x  =  y ) )
12 sbequ1 1871 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
1312com12 27 . . . 4  |-  ( ph  ->  ( x  =  y  ->  [ y  /  x ] ph ) )
149, 11, 13ee22 1352 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  ( ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
157, 14exlimexi 28586 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )
166, 15impbii 180 1  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531   [wsb 1638
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-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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