Theorem pm11.57 36739

Description: Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.57	|- ( A. x ph <-> A. x A. y ( ph /\ [ y / x ] ph ) )

Distinct variable group:   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem pm11.57

Step	Hyp	Ref	Expression
1		nfv 1761	. . . . 5 |- F/ y ph
2	1	nfal 2030	. . . 4 |- F/ y A. x ph
3		sp 1937	. . . . 5 |- ( A. x ph -> ph )
4		stdpc4 2184	. . . . 5 |- ( A. x ph -> [ y / x ] ph )
5	3, 4	jca 535	. . . 4 |- ( A. x ph -> ( ph /\ [ y / x ] ph ) )
6	2, 5	alrimi 1955	. . 3 |- ( A. x ph -> A. y ( ph /\ [ y / x ] ph ) )
7	6	axc4i 1980	. 2 |- ( A. x ph -> A. x A. y ( ph /\ [ y / x ] ph ) )
8		simpl 459	. . . 4 |- ( ( ph /\ [ y / x ] ph ) -> ph )
9	8	sps 1943	. . 3 |- ( A. y ( ph /\ [ y / x ] ph ) -> ph )
10	9	alimi 1684	. 2 |- ( A. x A. y ( ph /\ [ y / x ] ph ) -> A. x ph )
11	7, 10	impbii 191	1 |- ( A. x ph <-> A. x A. y ( ph /\ [ y / x ] ph ) )

Syntax hints:   <-> wb 188   /\ wa 371   A.wal 1442   [wsb 1797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798

This theorem is referenced by: (None)