Theorem sb8euh 2022

Description: Variable substitution in unique existential 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 1506	. . . . 5 |- ( ( ph <-> x = z ) -> A. w ( ph <-> x = z ) )
2	1	sb8h 1826	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 1932	. . . . . 6 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8euh.1	. . . . . . . 8 |- ( ph -> A. y ph )
5	4	hbsb 1922	. . . . . . 7 |- ( [ w / x ] ph -> A. y [ w / x ] ph )
6		equsb3 1924	. . . . . . . 8 |- ( [ w / x ] x = z <-> w = z )
7		ax-17 1506	. . . . . . . 8 |- ( w = z -> A. y w = z )
8	6, 7	hbxfrbi 1448	. . . . . . 7 |- ( [ w / x ] x = z -> A. y [ w / x ] x = z )
9	5, 8	hbbi 1527	. . . . . 6 |- ( ( [ w / x ] ph <-> [ w / x ] x = z ) -> A. y ( [ w / x ] ph <-> [ w / x ] x = z ) )
10	3, 9	hbxfrbi 1448	. . . . 5 |- ( [ w / x ] ( ph <-> x = z ) -> A. y [ w / x ] ( ph <-> x = z ) )
11		ax-17 1506	. . . . 5 |- ( [ y / x ] ( ph <-> x = z ) -> A. w [ y / x ] ( ph <-> x = z ) )
12		sbequ 1812	. . . . 5 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbvalh 1726	. . . 4 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 1924	. . . . . 6 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 1933	. . . . 5 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1446	. . . 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 1584	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
19		df-eu 2002	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
20		df-eu 2002	. 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:   -> wi 4   <-> wb 104   A.wal 1329   E.wex 1468   [wsb 1735   E!weu 1999

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002

This theorem is referenced by:  eu1  2024