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Theorem stdpc4 1896
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 2933 and ra4sbc 2999. (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 1888 . 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 1882
This theorem is referenced by:  sbft  1897  a4sbe  1967  a4sbim  1968  a4sbbi  1969  sb8  1986  sb9i  1988  pm13.183  2845  a4sbc  2933  nd1  8089  nd2  8090  pm10.14  26720  stdpc4-2  26735  pm11.57  26754
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 1883
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