Theorem pm11.59 36593

Description: Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
pm11.59	|- ( A. x ( ph -> ps ) -> A. y A. x ( ( ph /\ [ y / x ] ph ) -> ( ps /\ [ y / x ] ps ) ) )

Distinct variable groups:   ph, y   ps, y

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem pm11.59

Step	Hyp	Ref	Expression
1		nfv 1751	. . 3 |- F/ y ( ph -> ps )
2	1	nfal 2002	. 2 |- F/ y A. x ( ph -> ps )
3		sp 1909	. . . 4 |- ( A. x ( ph -> ps ) -> ( ph -> ps ) )
4		spsbim 2186	. . . 4 |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
5	3, 4	anim12d 565	. . 3 |- ( A. x ( ph -> ps ) -> ( ( ph /\ [ y / x ] ph ) -> ( ps /\ [ y / x ] ps ) ) )
6	5	axc4i 1952	. 2 |- ( A. x ( ph -> ps ) -> A. x ( ( ph /\ [ y / x ] ph ) -> ( ps /\ [ y / x ] ps ) ) )
7	2, 6	alrimi 1927	1 |- ( A. x ( ph -> ps ) -> A. y A. x ( ( ph /\ [ y / x ] ph ) -> ( ps /\ [ y / x ] ps ) ) )

Syntax hints:   -> wi 4   /\ wa 370   A.wal 1435   [wsb 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787

This theorem is referenced by: (None)