MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsb2 Structured version   Unicode version

Theorem dfsb2 2136
Description: An alternate definition of proper substitution that, like df-sb 1762, 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 1881 . . . 4  |-  ( A. x  x  =  y  ->  x  =  y )
2 sbequ2 1763 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
32sps 1887 . . . 4  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  ph )
)
4 orc 383 . . . 4  |-  ( ( x  =  y  /\  ph )  ->  ( (
x  =  y  /\  ph )  \/  A. x
( x  =  y  ->  ph ) ) )
51, 3, 4syl6an 543 . . 3  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  (
( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
) ) )
6 sb4 2119 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
7 olc 382 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )
86, 7syl6 31 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  ( ( x  =  y  /\  ph )  \/  A. x
( x  =  y  ->  ph ) ) ) )
95, 8pm2.61i 164 . 2  |-  ( [ y  /  x ] ph  ->  ( ( x  =  y  /\  ph )  \/  A. x
( x  =  y  ->  ph ) ) )
10 sbequ1 2017 . . . 4  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
1110imp 427 . . 3  |-  ( ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )
12 sb2 2115 . . 3  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
1311, 12jaoi 377 . 2  |-  ( ( ( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
)  ->  [ y  /  x ] ph )
149, 13impbii 188 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 184    \/ wo 366    /\ wa 367   A.wal 1401   [wsb 1761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-12 1876  ax-13 2024
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1632  df-nf 1636  df-sb 1762
This theorem is referenced by:  dfsb3  2137
  Copyright terms: Public domain W3C validator