Theorem sb8euh 1964

Description: Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)

Hypothesis

Ref	Expression
sb8euh.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
sb8euh	|- ( E! x ph <-> E! y [ y / x ] ph )

Proof of Theorem sb8euh

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

Step	Hyp	Ref	Expression
1		ax-17 1459	. . . . 5 |- ( ( ph <-> x = z ) -> A. w ( ph <-> x = z ) )
2	1	sb8h 1775	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 1874	. . . . . 6 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8euh.1	. . . . . . . 8 |- ( ph -> A. y ph )
5	4	hbsb 1864	. . . . . . 7 |- ( [ w / x ] ph -> A. y [ w / x ] ph )
6		equsb3 1866	. . . . . . . 8 |- ( [ w / x ] x = z <-> w = z )
7		ax-17 1459	. . . . . . . 8 |- ( w = z -> A. y w = z )
8	6, 7	hbxfrbi 1401	. . . . . . 7 |- ( [ w / x ] x = z -> A. y [ w / x ] x = z )
9	5, 8	hbbi 1480	. . . . . 6 |- ( ( [ w / x ] ph <-> [ w / x ] x = z ) -> A. y ( [ w / x ] ph <-> [ w / x ] x = z ) )
10	3, 9	hbxfrbi 1401	. . . . 5 |- ( [ w / x ] ( ph <-> x = z ) -> A. y [ w / x ] ( ph <-> x = z ) )
11		ax-17 1459	. . . . 5 |- ( [ y / x ] ( ph <-> x = z ) -> A. w [ y / x ] ( ph <-> x = z ) )
12		sbequ 1761	. . . . 5 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbvalh 1676	. . . 4 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 1866	. . . . . 6 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 1875	. . . . 5 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1399	. . . 4 |- ( A. y [ y / x ] ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
17	2, 13, 16	3bitri 204	. . 3 |- ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
18	17	exbii 1536	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
19		df-eu 1944	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
20		df-eu 1944	. 2 |- ( E! y [ y / x ] ph <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
21	18, 19, 20	3bitr4i 210	1 |- ( E! x ph <-> E! y [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   E.wex 1421   [wsb 1685   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944

This theorem is referenced by:  eu1  1966