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Theorem stdpc4 1700
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. (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 1385 . 2  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
3 sb2 1692 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
42, 3syl 14 1  |-  ( A. x ph  ->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1688
This theorem is referenced by:  sbh  1701  sbft  1771  pm13.183  2742  spsbc  2837
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