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Theorem stdpc4 1897
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 a4sbc 2947 and ra4sbc 3013. (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 1546 . 2  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
3 sb2 1889 . 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 1532   [wsb 1883
This theorem is referenced by:  sbft  1898  a4sbe  1968  a4sbim  1969  a4sbbi  1970  sb8  1987  sb9i  1989  pm13.183  2859  a4sbc  2947  nd1  8142  nd2  8143  pm10.14  26886  stdpc4-2  26901  pm11.57  26920
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-sb 1884
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