Theorem spsbim 2243

Description: Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
spsbim	|- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )

Proof of Theorem spsbim

Step	Hyp	Ref	Expression
1		stdpc4 2202	. 2 |- ( A. x ( ph -> ps ) -> [ y / x ] ( ph -> ps ) )
2		sbi1 2241	. 2 |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
3	1, 2	syl 17	1 |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   A.wal 1450   [wsb 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806

This theorem is referenced by:  mo3  2356  bj-hbsb3t  31379  wl-mo3t  31975  pm11.59  36811  sbiota1  36855