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Theorem sb2 2201
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2278) or a non-freeness hypothesis (sb6f 2234). (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 1957 . 2  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
2 equs4 2140 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
3 df-sb 1806 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
41, 2, 3sylanbrc 677 1  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> 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-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-sb 1806
This theorem is referenced by:  stdpc4  2202  sb3  2204  sb4b  2207  hbsb2  2208  hbsb2a  2210  hbsb2e  2212  equsb1  2217  equsb2  2218  dfsb2  2222  sbequi  2224  sb6f  2234  sbi1  2241  sb6  2278  iota4  5571  wl-lem-moexsb  31967  sbeqal1  36818
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