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Theorem sb4 2205
Description: One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 14-May-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 1808 . 2  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 equs5 2200 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) ) )
31, 2syl5ib 227 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    /\ wa 376   A.wal 1450   E.wex 1671   [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:  sb4b  2207  hbsb2  2208  dfsb2  2222  sbequi  2224  sbi1  2241
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