Theorem sb8eu 2027

Description: Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

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 1516	. . . . 5 |- F/ w ( ph <-> x = z )
2	1	sb8 1844	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 1947	. . . . . 6 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8eu.1	. . . . . . . 8 |- F/ y ph
5	4	nfsb 1934	. . . . . . 7 |- F/ y [ w / x ] ph
6		equsb3 1939	. . . . . . . 8 |- ( [ w / x ] x = z <-> w = z )
7		nfv 1516	. . . . . . . 8 |- F/ y w = z
8	6, 7	nfxfr 1462	. . . . . . 7 |- F/ y [ w / x ] x = z
9	5, 8	nfbi 1577	. . . . . 6 |- F/ y ( [ w / x ] ph <-> [ w / x ] x = z )
10	3, 9	nfxfr 1462	. . . . 5 |- F/ y [ w / x ] ( ph <-> x = z )
11		nfv 1516	. . . . 5 |- F/ w [ y / x ] ( ph <-> x = z )
12		sbequ 1828	. . . . 5 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbval 1742	. . . 4 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 1939	. . . . . 6 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 1948	. . . . 5 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1458	. . . 4 |- ( A. y [ y / x ] ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
17	2, 13, 16	3bitri 205	. . 3 |- ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
18	17	exbii 1593	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
19		df-eu 2017	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
20		df-eu 2017	. 2 |- ( E! y [ y / x ] ph <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
21	18, 19, 20	3bitr4i 211	1 |- ( E! x ph <-> E! y [ y / x ] ph )

Syntax hints:   <-> wb 104   A.wal 1341   F/wnf 1448   E.wex 1480   [wsb 1750   E!weu 2014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017

This theorem is referenced by:  sb8mo  2028  nfeud  2030  nfeu  2033  cbveu  2038  cbvreu  2690  acexmid  5841