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Theorem stdpc4 1971
Description: The specialization axiom of standard predicate calculus. It states that if a statement  ph holds for all  x, then it also holds for the specific case of  y (properly) substituted for  x. Translated to traditional notation, it can be read: " A. x ph ( x )  ->  ph ( y ), provided that  y is free for  x in  ph (
x )." Axiom 4 of [Mendelson] p. 69. See also spsbc 3006 and rspsbc 3072. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
stdpc4  |-  ( A. x ph  ->  [ y  /  x ] ph )

Proof of Theorem stdpc4
StepHypRef Expression
1 ax-1 7 . . 3  |-  ( ph  ->  ( x  =  y  ->  ph ) )
21alimi 1548 . 2  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
3 sb2 1970 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
42, 3syl 17 1  |-  ( A. x ph  ->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1529   [wsb 1632
This theorem is referenced by:  sbft  1972  spsbe  2016  spsbim  2017  spsbbi  2018  sb8  2033  sb9i  2035  pm13.183  2911  spsbc  3006  nd1  8206  nd2  8207  pm10.14  26955  stdpc4-2  26970  pm11.57  26989
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1531  df-sb 1633
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