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Theorem dfsb2 2212
Description: An alternate definition of proper substitution that, like df-sb 1808, 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 1947 . . . 4  |-  ( A. x  x  =  y  ->  x  =  y )
2 sbequ2 1809 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
32sps 1953 . . . 4  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  ph )
)
4 orc 391 . . . 4  |-  ( ( x  =  y  /\  ph )  ->  ( (
x  =  y  /\  ph )  \/  A. x
( x  =  y  ->  ph ) ) )
51, 3, 4syl6an 552 . . 3  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  (
( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
) ) )
6 sb4 2197 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
7 olc 390 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )
86, 7syl6 34 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  ( ( x  =  y  /\  ph )  \/  A. x
( x  =  y  ->  ph ) ) ) )
95, 8pm2.61i 169 . 2  |-  ( [ y  /  x ] ph  ->  ( ( x  =  y  /\  ph )  \/  A. x
( x  =  y  ->  ph ) ) )
10 sbequ1 2092 . . . 4  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
1110imp 435 . . 3  |-  ( ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )
12 sb2 2193 . . 3  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
1311, 12jaoi 385 . 2  |-  ( ( ( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
)  ->  [ y  /  x ] ph )
149, 13impbii 192 1  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375   A.wal 1452   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808
This theorem is referenced by:  dfsb3  2213
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