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Theorem sb4 1995
Description: One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sb4  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem sb4
StepHypRef Expression
1 sb1 1634 . 2  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 equs5 1938 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )
31, 2syl5 28 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    /\ wa 358   A.wal 1529   E.wex 1530   [wsb 1631
This theorem is referenced by:  sb4b  1996  dfsb2  1997  hbsb2  1999  sbn  2004  sbi1  2005  sbal1  2067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632
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