MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsb3 Unicode version

Theorem dfsb3 2109
Description: An alternate definition of proper substitution df-sb 1656 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)
Assertion
Ref Expression
dfsb3  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem dfsb3
StepHypRef Expression
1 df-or 360 . 2  |-  ( ( ( x  =  y  /\  ph )  \/ 
A. x ( x  =  y  ->  ph )
)  <->  ( -.  (
x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
2 dfsb2 2108 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  /\  ph )  \/  A. x ( x  =  y  ->  ph )
) )
3 imnan 412 . . 3  |-  ( ( x  =  y  ->  -.  ph )  <->  -.  (
x  =  y  /\  ph ) )
43imbi1i 316 . 2  |-  ( ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph ) )  <-> 
( -.  ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )
51, 2, 43bitr4i 269 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 177    \/ wo 358    /\ wa 359   A.wal 1546   [wsb 1655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
  Copyright terms: Public domain W3C validator