Theorem wl-sb8eut 33359

Description: Substitution of variable in universal quantifier. Closed form of sb8eu 2503. (Contributed by Wolf Lammen, 11-Aug-2019.)

Assertion

Ref	Expression
wl-sb8eut	|- ( A. x F/ y ph -> ( E! x ph <-> E! y [ y / x ] ph ) )

Proof of Theorem wl-sb8eut

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnf1 2031	. . . . . 6 |- F/ y F/ y ph
2	1	nfal 2153	. . . . 5 |- F/ y A. x F/ y ph
3		equsb3 2432	. . . . . . 7 |- ( [ v / x ] x = u <-> v = u )
4	3	sblbis 2404	. . . . . 6 |- ( [ v / x ] ( ph <-> x = u ) <-> ( [ v / x ] ph <-> v = u ) )
5		nfa1 2028	. . . . . . . 8 |- F/ x A. x F/ y ph
6		sp 2053	. . . . . . . 8 |- ( A. x F/ y ph -> F/ y ph )
7	5, 6	nfsbd 2442	. . . . . . 7 |- ( A. x F/ y ph -> F/ y [ v / x ] ph )
8		nfvd 1844	. . . . . . 7 |- ( A. x F/ y ph -> F/ y v = u )
9	7, 8	nfbid 1832	. . . . . 6 |- ( A. x F/ y ph -> F/ y ( [ v / x ] ph <-> v = u ) )
10	4, 9	nfxfrd 1780	. . . . 5 |- ( A. x F/ y ph -> F/ y [ v / x ] ( ph <-> x = u ) )
11		sbequ 2376	. . . . . 6 |- ( v = y -> ( [ v / x ] ( ph <-> x = u ) <-> [ y / x ] ( ph <-> x = u ) ) )
12	11	a1i 11	. . . . 5 |- ( A. x F/ y ph -> ( v = y -> ( [ v / x ] ( ph <-> x = u ) <-> [ y / x ] ( ph <-> x = u ) ) ) )
13	2, 10, 12	cbvald 2277	. . . 4 |- ( A. x F/ y ph -> ( A. v [ v / x ] ( ph <-> x = u ) <-> A. y [ y / x ] ( ph <-> x = u ) ) )
14		nfv 1843	. . . . . 6 |- F/ v ( ph <-> x = u )
15	14	sb8 2424	. . . . 5 |- ( A. x ( ph <-> x = u ) <-> A. v [ v / x ] ( ph <-> x = u ) )
16	15	bicomi 214	. . . 4 |- ( A. v [ v / x ] ( ph <-> x = u ) <-> A. x ( ph <-> x = u ) )
17		equsb3 2432	. . . . . 6 |- ( [ y / x ] x = u <-> y = u )
18	17	sblbis 2404	. . . . 5 |- ( [ y / x ] ( ph <-> x = u ) <-> ( [ y / x ] ph <-> y = u ) )
19	18	albii 1747	. . . 4 |- ( A. y [ y / x ] ( ph <-> x = u ) <-> A. y ( [ y / x ] ph <-> y = u ) )
20	13, 16, 19	3bitr3g 302	. . 3 |- ( A. x F/ y ph -> ( A. x ( ph <-> x = u ) <-> A. y ( [ y / x ] ph <-> y = u ) ) )
21	20	exbidv 1850	. 2 |- ( A. x F/ y ph -> ( E. u A. x ( ph <-> x = u ) <-> E. u A. y ( [ y / x ] ph <-> y = u ) ) )
22		df-eu 2474	. 2 |- ( E! x ph <-> E. u A. x ( ph <-> x = u ) )
23		df-eu 2474	. 2 |- ( E! y [ y / x ] ph <-> E. u A. y ( [ y / x ] ph <-> y = u ) )
24	21, 22, 23	3bitr4g 303	1 |- ( A. x F/ y ph -> ( E! x ph <-> E! y [ y / x ] ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708   [wsb 1880   E!weu 2470

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474

This theorem is referenced by:  wl-sb8mot  33360