Theorem sb8eu 2051

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 1539	. . . . 5 |- F/ w ( ph <-> x = z )
2	1	sb8 1867	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 1971	. . . . . 6 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8eu.1	. . . . . . . 8 |- F/ y ph
5	4	nfsb 1958	. . . . . . 7 |- F/ y [ w / x ] ph
6		equsb3 1963	. . . . . . . 8 |- ( [ w / x ] x = z <-> w = z )
7		nfv 1539	. . . . . . . 8 |- F/ y w = z
8	6, 7	nfxfr 1485	. . . . . . 7 |- F/ y [ w / x ] x = z
9	5, 8	nfbi 1600	. . . . . 6 |- F/ y ( [ w / x ] ph <-> [ w / x ] x = z )
10	3, 9	nfxfr 1485	. . . . 5 |- F/ y [ w / x ] ( ph <-> x = z )
11		nfv 1539	. . . . 5 |- F/ w [ y / x ] ( ph <-> x = z )
12		sbequ 1851	. . . . 5 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbval 1765	. . . 4 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 1963	. . . . . 6 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 1972	. . . . 5 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1481	. . . 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 1616	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
19		df-eu 2041	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
20		df-eu 2041	. 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 1471   E.wex 1503   [wsb 1773   E!weu 2038

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041

This theorem is referenced by:  sb8mo  2052  nfeud  2054  nfeu  2057  cbveu  2062  cbvreu  2716  acexmid  5891