Theorem sbnf2 2439

Description: Two ways of expressing " x is (effectively) not free in ph." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.)

Assertion

Ref	Expression
sbnf2	|- ( F/ x ph <-> A. y A. z ( [ y / x ] ph <-> [ z / x ] ph ) )

Distinct variable groups:   x, y, z   ph, y, z

Allowed substitution hint:   ph( x)

Proof of Theorem sbnf2

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . . 6 |- F/ y ph
2	1	sb8e 2425	. . . . 5 |- ( E. x ph <-> E. y [ y / x ] ph )
3		nfv 1843	. . . . . 6 |- F/ z ph
4	3	sb8 2424	. . . . 5 |- ( A. x ph <-> A. z [ z / x ] ph )
5	2, 4	imbi12i 340	. . . 4 |- ( ( E. x ph -> A. x ph ) <-> ( E. y [ y / x ] ph -> A. z [ z / x ] ph ) )
6		df-nf 1710	. . . 4 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
7		pm11.53v 1906	. . . 4 |- ( A. y A. z ( [ y / x ] ph -> [ z / x ] ph ) <-> ( E. y [ y / x ] ph -> A. z [ z / x ] ph ) )
8	5, 6, 7	3bitr4i 292	. . 3 |- ( F/ x ph <-> A. y A. z ( [ y / x ] ph -> [ z / x ] ph ) )
9	3	sb8e 2425	. . . . . 6 |- ( E. x ph <-> E. z [ z / x ] ph )
10	1	sb8 2424	. . . . . 6 |- ( A. x ph <-> A. y [ y / x ] ph )
11	9, 10	imbi12i 340	. . . . 5 |- ( ( E. x ph -> A. x ph ) <-> ( E. z [ z / x ] ph -> A. y [ y / x ] ph ) )
12		pm11.53v 1906	. . . . 5 |- ( A. z A. y ( [ z / x ] ph -> [ y / x ] ph ) <-> ( E. z [ z / x ] ph -> A. y [ y / x ] ph ) )
13	11, 12	bitr4i 267	. . . 4 |- ( ( E. x ph -> A. x ph ) <-> A. z A. y ( [ z / x ] ph -> [ y / x ] ph ) )
14		alcom 2037	. . . 4 |- ( A. z A. y ( [ z / x ] ph -> [ y / x ] ph ) <-> A. y A. z ( [ z / x ] ph -> [ y / x ] ph ) )
15	6, 13, 14	3bitri 286	. . 3 |- ( F/ x ph <-> A. y A. z ( [ z / x ] ph -> [ y / x ] ph ) )
16	8, 15	anbi12i 733	. 2 |- ( ( F/ x ph /\ F/ x ph ) <-> ( A. y A. z ( [ y / x ] ph -> [ z / x ] ph ) /\ A. y A. z ( [ z / x ] ph -> [ y / x ] ph ) ) )
17		pm4.24 675	. 2 |- ( F/ x ph <-> ( F/ x ph /\ F/ x ph ) )
18		2albiim 1817	. 2 |- ( A. y A. z ( [ y / x ] ph <-> [ z / x ] ph ) <-> ( A. y A. z ( [ y / x ] ph -> [ z / x ] ph ) /\ A. y A. z ( [ z / x ] ph -> [ y / x ] ph ) ) )
19	16, 17, 18	3bitr4i 292	1 |- ( F/ x ph <-> A. y A. z ( [ y / x ] ph <-> [ z / x ] ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   [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:  sbnfc2  4007  nfnid  4897