Theorem pm11.57 31539

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 1712	. . . . 5 |- F/ y ph
2	1	nfal 1952	. . . 4 |- F/ y A. x ph
3		sp 1864	. . . . 5 |- ( A. x ph -> ph )
4		stdpc4 2096	. . . . 5 |- ( A. x ph -> [ y / x ] ph )
5	3, 4	jca 530	. . . 4 |- ( A. x ph -> ( ph /\ [ y / x ] ph ) )
6	2, 5	alrimi 1882	. . 3 |- ( A. x ph -> A. y ( ph /\ [ y / x ] ph ) )
7	6	axc4i 1903	. 2 |- ( A. x ph -> A. x A. y ( ph /\ [ y / x ] ph ) )
8		simpl 455	. . . 4 |- ( ( ph /\ [ y / x ] ph ) -> ph )
9	8	sps 1870	. . 3 |- ( A. y ( ph /\ [ y / x ] ph ) -> ph )
10	9	alimi 1638	. 2 |- ( A. x A. y ( ph /\ [ y / x ] ph ) -> A. x ph )
11	7, 10	impbii 188	1 |- ( A. x ph <-> A. x A. y ( ph /\ [ y / x ] ph ) )

Syntax hints:   <-> wb 184   /\ wa 367   A.wal 1396   [wsb 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745

This theorem is referenced by: (None)