Theorem sb8eu 2058

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 1542	. . . . 5 |- F/ w ( ph <-> x = z )
2	1	sb8 1870	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 1978	. . . . . 6 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8eu.1	. . . . . . . 8 |- F/ y ph
5	4	nfsb 1965	. . . . . . 7 |- F/ y [ w / x ] ph
6		equsb3 1970	. . . . . . . 8 |- ( [ w / x ] x = z <-> w = z )
7		nfv 1542	. . . . . . . 8 |- F/ y w = z
8	6, 7	nfxfr 1488	. . . . . . 7 |- F/ y [ w / x ] x = z
9	5, 8	nfbi 1603	. . . . . 6 |- F/ y ( [ w / x ] ph <-> [ w / x ] x = z )
10	3, 9	nfxfr 1488	. . . . 5 |- F/ y [ w / x ] ( ph <-> x = z )
11		nfv 1542	. . . . 5 |- F/ w [ y / x ] ( ph <-> x = z )
12		sbequ 1854	. . . . 5 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbval 1768	. . . 4 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 1970	. . . . . 6 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 1979	. . . . 5 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1484	. . . 4 |- ( A. y [ y / x ] ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
17	2, 13, 16	3bitri 206	. . 3 |- ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
18	17	exbii 1619	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
19		df-eu 2048	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
20		df-eu 2048	. 2 |- ( E! y [ y / x ] ph <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
21	18, 19, 20	3bitr4i 212	1 |- ( E! x ph <-> E! y [ y / x ] ph )

Syntax hints:   <-> wb 105   A.wal 1362   F/wnf 1474   E.wex 1506   [wsb 1776   E!weu 2045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048

This theorem is referenced by:  sb8mo  2059  nfeud  2061  nfeu  2064  cbveu  2069  cbvreu  2727  acexmid  5921