Theorem eu1 1998

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 1887	. . 3 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
2	1	euf 1978	. 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 1996	. 2 |- ( E! x ph <-> E! y [ y / x ] ph )
5		equcom 1663	. . . . . . 7 |- ( x = y <-> y = x )
6	5	imbi2i 225	. . . . . 6 |- ( ( [ y / x ] ph -> x = y ) <-> ( [ y / x ] ph -> y = x ) )
7	6	albii 1427	. . . . 5 |- ( A. y ( [ y / x ] ph -> x = y ) <-> A. y ( [ y / x ] ph -> y = x ) )
8	3	sb6rf 1805	. . . . 5 |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) )
9	7, 8	anbi12i 453	. . . 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 1444	. . . 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 1565	. 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 1310   E.wex 1449   [wsb 1716   E!weu 1973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-eu 1976

This theorem is referenced by:  euex  2003  eu2  2017