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Theorem sb5ALT 28546
Description: Equivalence for substitution. Alternate proof of sb5 2175. This proof is sb5ALTVD 28962 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 2113 . . . 4  |-  [ y  /  x ] x  =  y
2 sban 2143 . . . . 5  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  <->  ( [
y  /  x ]
x  =  y  /\  [ y  /  x ] ph ) )
32simplbi2com 1383 . . . 4  |-  ( [ y  /  x ] ph  ->  ( [ y  /  x ] x  =  y  ->  [ y  /  x ] ( x  =  y  /\  ph ) ) )
41, 3mpi 17 . . 3  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ( x  =  y  /\  ph )
)
5 spsbe 1663 . . 3  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  ph )
)
64, 5syl 16 . 2  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
7 hbs1 2180 . . 3  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
8 simpr 448 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  ph )
98a1i 11 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  ( ( x  =  y  /\  ph )  ->  ph ) )
10 simpl 444 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  x  =  y )
1110a1i 11 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  ( ( x  =  y  /\  ph )  ->  x  =  y ) )
12 sbequ1 1943 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
1312com12 29 . . . 4  |-  ( ph  ->  ( x  =  y  ->  [ y  /  x ] ph ) )
149, 11, 13ee22 1371 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  ( ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
157, 14exlimexi 28545 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )
166, 15impbii 181 1  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550   [wsb 1658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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