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Theorem dfsb2 2008
Description: An alternate definition of proper substitution that, like df-sb 1639, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)
Assertion
Ref Expression
dfsb2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem dfsb2
StepHypRef Expression
1 sp 1728 . . . 4  |-  ( A. x  x  =  y  ->  x  =  y )
2 sbequ2 1640 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
32sps 1751 . . . 4  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  ph )
)
4 orc 374 . . . 4  |-  ( ( x  =  y  /\  ph )  ->  ( (
x  =  y  /\  ph )  \/  A. x
( x  =  y  ->  ph ) ) )
51, 3, 4ee12an 1353 . . 3  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  (
( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
) ) )
6 sb4 2006 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
7 olc 373 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )
86, 7syl6 29 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  ( ( x  =  y  /\  ph )  \/  A. x
( x  =  y  ->  ph ) ) ) )
95, 8pm2.61i 156 . 2  |-  ( [ y  /  x ] ph  ->  ( ( x  =  y  /\  ph )  \/  A. x
( x  =  y  ->  ph ) ) )
10 sbequ1 1871 . . . 4  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
1110imp 418 . . 3  |-  ( ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )
12 sb2 1976 . . 3  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
1311, 12jaoi 368 . 2  |-  ( ( ( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
)  ->  [ y  /  x ] ph )
149, 13impbii 180 1  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1530   [wsb 1638
This theorem is referenced by:  dfsb3  2009
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-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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