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Theorem stdpc4 2148
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 3313 and rspsbc 3379. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
stdpc4  |-  ( A. x ph  ->  [ y  /  x ] ph )

Proof of Theorem stdpc4
StepHypRef Expression
1 ala1 1706 . 2  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
2 sb2 2147 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
31, 2syl 17 1  |-  ( A. x ph  ->  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1436   [wsb 1787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-12 1906  ax-13 2054
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1661  df-sb 1788
This theorem is referenced by:  2stdpc4  2149  sbft  2174  spsbim  2189  spsbbi  2197  sbt  2214  sbtrt  2215  pm13.183  3213  spsbc  3313  nd1  9014  nd2  9015  bj-vexwt  31427  axfrege58b  36360  pm10.14  36572  pm11.57  36603
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