Theorem sb8eu 2352

Description: Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.)

Hypothesis

Ref	Expression
sb8eu.1	|- F/ y ph

Assertion

Ref	Expression
sb8eu	|- ( E! x ph <-> E! y [ y / x ] ph )

Proof of Theorem sb8eu

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1769	. . . . 5 |- F/ w ( ph <-> x = z )
2	1	sb8 2273	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		equsb3 2281	. . . . . 6 |- ( [ w / x ] x = z <-> w = z )
4	3	sblbis 2253	. . . . 5 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> w = z ) )
5	4	albii 1699	. . . 4 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. w ( [ w / x ] ph <-> w = z ) )
6		sb8eu.1	. . . . . . 7 |- F/ y ph
7	6	nfsb 2289	. . . . . 6 |- F/ y [ w / x ] ph
8		nfv 1769	. . . . . 6 |- F/ y w = z
9	7, 8	nfbi 2037	. . . . 5 |- F/ y ( [ w / x ] ph <-> w = z )
10		nfv 1769	. . . . 5 |- F/ w ( [ y / x ] ph <-> y = z )
11		sbequ 2225	. . . . . 6 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
12		equequ1 1875	. . . . . 6 |- ( w = y -> ( w = z <-> y = z ) )
13	11, 12	bibi12d 328	. . . . 5 |- ( w = y -> ( ( [ w / x ] ph <-> w = z ) <-> ( [ y / x ] ph <-> y = z ) ) )
14	9, 10, 13	cbval 2127	. . . 4 |- ( A. w ( [ w / x ] ph <-> w = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
15	2, 5, 14	3bitri 279	. . 3 |- ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
16	15	exbii 1726	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
17		df-eu 2323	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
18		df-eu 2323	. 2 |- ( E! y [ y / x ] ph <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
19	16, 17, 18	3bitr4i 285	1 |- ( E! x ph <-> E! y [ y / x ] ph )

Syntax hints:   <-> wb 189   A.wal 1450   E.wex 1671   F/wnf 1675   [wsb 1805   E!weu 2319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323

This theorem is referenced by:  sb8mo  2353  cbveu  2354  eu1  2359  cbvreu  3003