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Theorem sb4b 1994
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 1993 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
2 sb2 1963 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
31, 2impbid1 194 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 176   A.wal 1527   [wsb 1629
This theorem is referenced by:  nfsb4t  2020  sbcom  2029  sbcom2  2053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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