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Theorem sb2 2079
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2159) or a non-freeness hypothesis (sb6f 2112). (Contributed by NM, 13-May-1993.)
Assertion
Ref Expression
sb2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )

Proof of Theorem sb2
StepHypRef Expression
1 sp 1845 . 2  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
2 equs4 2021 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
3 df-sb 1727 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
41, 2, 3sylanbrc 664 1  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1381   E.wex 1599   [wsb 1726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-12 1840  ax-13 1985
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1600  df-sb 1727
This theorem is referenced by:  stdpc4  2080  sb3  2082  sb4b  2084  hbsb2  2085  hbsb2a  2087  hbsb2e  2088  equsb1  2093  equsb2  2094  dfsb2  2100  sbequi  2102  sb6f  2112  sbi1  2119  sb6  2159  iota4  5559  wl-lem-moexsb  29993  sbeqal1  31258
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