MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb4b Unicode version

Theorem sb4b 2111
Description: Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
Assertion
Ref Expression
sb4b  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
) )

Proof of Theorem sb4b
StepHypRef Expression
1 sb4 2110 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
2 sb2 2079 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
31, 2impbid1 195 1  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1546   [wsb 1655
This theorem is referenced by:  nfsb4t  2137  sbcom  2146  sbcom2  2171
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-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
  Copyright terms: Public domain W3C validator