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Theorem sb3 2088
Description: One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sb3  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )

Proof of Theorem sb3
StepHypRef Expression
1 equs5 2085 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )
2 sb2 2086 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
31, 2syl6 31 1  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550   [wsb 1658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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