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Theorem stdpc4 2073
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 3133 and rspsbc 3199. (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 5 . . 3  |-  ( ph  ->  ( x  =  y  ->  ph ) )
21alimi 1565 . 2  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
3 sb2 2072 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
42, 3syl 16 1  |-  ( A. x ph  ->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546   [wsb 1655
This theorem is referenced by:  sbft  2074  spsbe  2124  spsbim  2125  spsbbi  2126  sb9i  2143  pm13.183  3036  spsbc  3133  nd1  8418  nd2  8419  pm10.14  27422  stdpc4-2  27437  pm11.57  27456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-sb 1656
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