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Theorem sb4 1992
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 1633 . 2  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 equs5 1939 . 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 1527   E.wex 1528   [wsb 1630
This theorem is referenced by:  sb4b  1993  dfsb2  1994  hbsb2  1996  sbn  2001  sbi1  2002  sbal1  2068
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 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631
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