Theorem eu1 2510

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.)

Hypothesis

Ref	Expression
eu1.1	|- F/ 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		nfs1v 2437	. . 3 |- F/ x [ y / x ] ph
2	1	euf 2478	. 2 |- ( E! y [ y / x ] ph <-> E. x A. y ( [ y / x ] ph <-> y = x ) )
3		eu1.1	. . 3 |- F/ y ph
4	3	sb8eu 2503	. 2 |- ( E! x ph <-> E! y [ y / x ] ph )
5	3	sb6rf 2423	. . . . 5 |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) )
6		equcom 1945	. . . . . . 7 |- ( x = y <-> y = x )
7	6	imbi2i 326	. . . . . 6 |- ( ( [ y / x ] ph -> x = y ) <-> ( [ y / x ] ph -> y = x ) )
8	7	albii 1747	. . . . 5 |- ( A. y ( [ y / x ] ph -> x = y ) <-> A. y ( [ y / x ] ph -> y = x ) )
9	5, 8	anbi12ci 734	. . . 4 |- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> ( A. y ( [ y / x ] ph -> y = x ) /\ A. y ( y = x -> [ y / x ] ph ) ) )
10		albiim 1816	. . . 4 |- ( A. y ( [ y / x ] ph <-> y = x ) <-> ( A. y ( [ y / x ] ph -> y = x ) /\ A. y ( y = x -> [ y / x ] ph ) ) )
11	9, 10	bitr4i 267	. . 3 |- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> A. y ( [ y / x ] ph <-> y = x ) )
12	11	exbii 1774	. 2 |- ( E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> E. x A. y ( [ y / x ] ph <-> y = x ) )
13	2, 4, 12	3bitr4i 292	1 |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   [wsb 1880   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474

This theorem is referenced by:  euexALT  2511  kmlem15  8986