Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  stdpc4 Unicode version

Theorem stdpc4 1964
 Description: The specialization axiom of standard predicate calculus. It states that if a statement holds for all , then it also holds for the specific case of (properly) substituted for . Translated to traditional notation, it can be read: " , provided that is free for in ." Axiom 4 of [Mendelson] p. 69. See also spsbc 3003 and rspsbc 3069. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
stdpc4

Proof of Theorem stdpc4
StepHypRef Expression
1 ax-1 5 . . 3
21alimi 1546 . 2
3 sb2 1963 . 2
42, 3syl 15 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1527  wsb 1629 This theorem is referenced by:  sbft  1965  spsbe  2015  spsbim  2016  spsbbi  2017  sb8  2032  sb9i  2034  pm13.183  2908  spsbc  3003  nd1  8209  nd2  8210  pm10.14  27554  stdpc4-2  27569  pm11.57  27588 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-sb 1630
 Copyright terms: Public domain W3C validator