Theorem stdpc4 1748

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

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 |- ( ph -> ( x = y -> ph ) )
2	1	alimi 1431	. 2 |- ( A. x ph -> A. x ( x = y -> ph ) )
3		sb2 1740	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
4	2, 3	syl 14	1 |- ( A. x ph -> [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1329   [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1736

This theorem is referenced by:  sbh  1749  sbft  1820  pm13.183  2817  spsbc  2915