Theorem pm11.57 38589

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 1843	. . . . 5 |- F/ y ph
2	1	nfal 2153	. . . 4 |- F/ y A. x ph
3		sp 2053	. . . . 5 |- ( A. x ph -> ph )
4		stdpc4 2353	. . . . 5 |- ( A. x ph -> [ y / x ] ph )
5	3, 4	jca 554	. . . 4 |- ( A. x ph -> ( ph /\ [ y / x ] ph ) )
6	2, 5	alrimi 2082	. . 3 |- ( A. x ph -> A. y ( ph /\ [ y / x ] ph ) )
7	6	axc4i 2131	. 2 |- ( A. x ph -> A. x A. y ( ph /\ [ y / x ] ph ) )
8		simpl 473	. . . 4 |- ( ( ph /\ [ y / x ] ph ) -> ph )
9	8	sps 2055	. . 3 |- ( A. y ( ph /\ [ y / x ] ph ) -> ph )
10	9	alimi 1739	. 2 |- ( A. x A. y ( ph /\ [ y / x ] ph ) -> A. x ph )
11	7, 10	impbii 199	1 |- ( A. x ph <-> A. x A. y ( ph /\ [ y / x ] ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)