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

Theorem sb4b 2207
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 2205 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
2 sb2 2201 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
31, 2impbid1 208 1  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1450   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by:  sbcom3  2260  sbal1  2309  sbal2  2310  wl-sb6nae  31956  wl-sbalnae  31962  wl-sbcom3  31989
  Copyright terms: Public domain W3C validator