Theorem eu1 1925

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 1814	. . 3 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
2	1	euf 1905	. 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 1923	. 2 |- ( E! x ph <-> E! y [ y / x ] ph )
5		equcom 1593	. . . . . . 7 |- ( x = y <-> y = x )
6	5	imbi2i 215	. . . . . 6 |- ( ( [ y / x ] ph -> x = y ) <-> ( [ y / x ] ph -> y = x ) )
7	6	albii 1359	. . . . 5 |- ( A. y ( [ y / x ] ph -> x = y ) <-> A. y ( [ y / x ] ph -> y = x ) )
8	3	sb6rf 1733	. . . . 5 |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) )
9	7, 8	anbi12i 433	. . . 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 253	. . . 4 |- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> ( A. y ( [ y / x ] ph -> x = y ) /\ ph ) )
11		albiim 1376	. . . 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 201	. . 3 |- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> A. y ( [ y / x ] ph <-> y = x ) )
13	12	exbii 1496	. 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 201	1 |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   [wsb 1645   E!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903

This theorem is referenced by:  euex  1930  eu2  1944