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Theorem sb4b 2098
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 2097 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
2 sb2 2093 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
31, 2impbid1 196 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 178   A.wal 1550   [wsb 1659
This theorem is referenced by:  nfsb4tOLD  2132  sbcom  2170  sbcomOLD  2171  sbcom2  2196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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