Theorem eu1 2022

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)

Hypothesis

Ref	Expression
eu1.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
eu1	|- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem eu1

Step	Hyp	Ref	Expression
1		hbs1 1909	. . 3 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
2	1	euf 2002	. 2 |- ( E! y [ y / x ] ph <-> E. x A. y ( [ y / x ] ph <-> y = x ) )
3		eu1.1	. . 3 |- ( ph -> A. y ph )
4	3	sb8euh 2020	. 2 |- ( E! x ph <-> E! y [ y / x ] ph )
5		equcom 1682	. . . . . . 7 |- ( x = y <-> y = x )
6	5	imbi2i 225	. . . . . 6 |- ( ( [ y / x ] ph -> x = y ) <-> ( [ y / x ] ph -> y = x ) )
7	6	albii 1446	. . . . 5 |- ( A. y ( [ y / x ] ph -> x = y ) <-> A. y ( [ y / x ] ph -> y = x ) )
8	3	sb6rf 1825	. . . . 5 |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) )
9	7, 8	anbi12i 455	. . . 4 |- ( ( A. y ( [ y / x ] ph -> x = y ) /\ ph ) <-> ( A. y ( [ y / x ] ph -> y = x ) /\ A. y ( y = x -> [ y / x ] ph ) ) )
10		ancom 264	. . . 4 |- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> ( A. y ( [ y / x ] ph -> x = y ) /\ ph ) )
11		albiim 1463	. . . 4 |- ( A. y ( [ y / x ] ph <-> y = x ) <-> ( A. y ( [ y / x ] ph -> y = x ) /\ A. y ( y = x -> [ y / x ] ph ) ) )
12	9, 10, 11	3bitr4i 211	. . 3 |- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> A. y ( [ y / x ] ph <-> y = x ) )
13	12	exbii 1584	. 2 |- ( E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> E. x A. y ( [ y / x ] ph <-> y = x ) )
14	2, 4, 13	3bitr4i 211	1 |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   E.wex 1468   [wsb 1735   E!weu 1997

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 2000

This theorem is referenced by:  euex  2027  eu2  2041