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Theorem sb5ALT 36519
Description: Equivalence for substitution. Alternate proof of sb5 2226. This proof is sb5ALTVD 36950 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 2161 . . . 4  |-  [ y  /  x ] x  =  y
2 sban 2194 . . . . 5  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  <->  ( [
y  /  x ]
x  =  y  /\  [ y  /  x ] ph ) )
32simplbi2com 631 . . . 4  |-  ( [ y  /  x ] ph  ->  ( [ y  /  x ] x  =  y  ->  [ y  /  x ] ( x  =  y  /\  ph ) ) )
41, 3mpi 21 . . 3  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ( x  =  y  /\  ph )
)
5 spsbe 1793 . . 3  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  ph )
)
64, 5syl 17 . 2  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
7 hbs1 2232 . . 3  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
8 simpr 462 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  ph )
98a1i 11 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  ( ( x  =  y  /\  ph )  ->  ph ) )
10 simpl 458 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  x  =  y )
1110a1i 11 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  ( ( x  =  y  /\  ph )  ->  x  =  y ) )
12 sbequ1 2048 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
1312com12 32 . . . 4  |-  ( ph  ->  ( x  =  y  ->  [ y  /  x ] ph ) )
149, 11, 13syl6c 66 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  ( ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
157, 14exlimexi 36518 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )
166, 15impbii 190 1  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   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-or 371  df-an 372  df-ex 1660  df-nf 1664  df-sb 1790
This theorem is referenced by: (None)
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