Theorem eu1 2080

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 1967	. . 3 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
2	1	euf 2060	. 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 2078	. 2 |- ( E! x ph <-> E! y [ y / x ] ph )
5		equcom 1730	. . . . . . 7 |- ( x = y <-> y = x )
6	5	imbi2i 226	. . . . . 6 |- ( ( [ y / x ] ph -> x = y ) <-> ( [ y / x ] ph -> y = x ) )
7	6	albii 1494	. . . . 5 |- ( A. y ( [ y / x ] ph -> x = y ) <-> A. y ( [ y / x ] ph -> y = x ) )
8	3	sb6rf 1877	. . . . 5 |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) )
9	7, 8	anbi12i 460	. . . 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 266	. . . 4 |- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> ( A. y ( [ y / x ] ph -> x = y ) /\ ph ) )
11		albiim 1511	. . . 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 212	. . 3 |- ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> A. y ( [ y / x ] ph <-> y = x ) )
13	12	exbii 1629	. 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 212	1 |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   E.wex 1516   [wsb 1786   E!weu 2055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058

This theorem is referenced by:  euex  2085  eu2  2100