Theorem mo5f 25813

Description: Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)

Hypotheses

Ref	Expression
mo5f.1	|- F/ i ph
mo5f.2	|- F/ j ph

Assertion

Ref	Expression
mo5f	|- ( E* x ph <-> A. i A. j ( ( [ i / x ] ph /\ [ j / x ] ph ) -> i = j ) )

Distinct variable group:   i, j, x

Allowed substitution hints:   ph( x, i, j)

Proof of Theorem mo5f

Step	Hyp	Ref	Expression
1		mo5f.2	. . 3 |- F/ j ph
2	1	mo3 2296	. 2 |- ( E* x ph <-> A. x A. j ( ( ph /\ [ j / x ] ph ) -> x = j ) )
3		mo5f.1	. . . . . 6 |- F/ i ph
4	3	nfsb 2146	. . . . . 6 |- F/ i [ j / x ] ph
5	3, 4	nfan 1860	. . . . 5 |- F/ i ( ph /\ [ j / x ] ph )
6		nfv 1673	. . . . 5 |- F/ i x = j
7	5, 6	nfim 1852	. . . 4 |- F/ i ( ( ph /\ [ j / x ] ph ) -> x = j )
8	7	nfal 1872	. . 3 |- F/ i A. j ( ( ph /\ [ j / x ] ph ) -> x = j )
9	8	sb8 2124	. 2 |- ( A. x A. j ( ( ph /\ [ j / x ] ph ) -> x = j ) <-> A. i [ i / x ] A. j ( ( ph /\ [ j / x ] ph ) -> x = j ) )
10		sbim 2087	. . . . 5 |- ( [ i / x ] ( ( ph /\ [ j / x ] ph ) -> x = j ) <-> ( [ i / x ] ( ph /\ [ j / x ] ph ) -> [ i / x ] x = j ) )
11		sban 2091	. . . . . . 7 |- ( [ i / x ] ( ph /\ [ j / x ] ph ) <-> ( [ i / x ] ph /\ [ i / x ] [ j / x ] ph ) )
12		nfs1v 2142	. . . . . . . . . 10 |- F/ x [ j / x ] ph
13	12	sbf 2071	. . . . . . . . 9 |- ( [ i / x ] [ j / x ] ph <-> [ j / x ] ph )
14	13	bicomi 202	. . . . . . . 8 |- ( [ j / x ] ph <-> [ i / x ] [ j / x ] ph )
15	14	anbi2i 694	. . . . . . 7 |- ( ( [ i / x ] ph /\ [ j / x ] ph ) <-> ( [ i / x ] ph /\ [ i / x ] [ j / x ] ph ) )
16	11, 15	bitr4i 252	. . . . . 6 |- ( [ i / x ] ( ph /\ [ j / x ] ph ) <-> ( [ i / x ] ph /\ [ j / x ] ph ) )
17		equsb3 2137	. . . . . 6 |- ( [ i / x ] x = j <-> i = j )
18	16, 17	imbi12i 326	. . . . 5 |- ( ( [ i / x ] ( ph /\ [ j / x ] ph ) -> [ i / x ] x = j ) <-> ( ( [ i / x ] ph /\ [ j / x ] ph ) -> i = j ) )
19	10, 18	bitri 249	. . . 4 |- ( [ i / x ] ( ( ph /\ [ j / x ] ph ) -> x = j ) <-> ( ( [ i / x ] ph /\ [ j / x ] ph ) -> i = j ) )
20	19	sbalv 2177	. . 3 |- ( [ i / x ] A. j ( ( ph /\ [ j / x ] ph ) -> x = j ) <-> A. j ( ( [ i / x ] ph /\ [ j / x ] ph ) -> i = j ) )
21	20	albii 1610	. 2 |- ( A. i [ i / x ] A. j ( ( ph /\ [ j / x ] ph ) -> x = j ) <-> A. i A. j ( ( [ i / x ] ph /\ [ j / x ] ph ) -> i = j ) )
22	2, 9, 21	3bitri 271	1 |- ( E* x ph <-> A. i A. j ( ( [ i / x ] ph /\ [ j / x ] ph ) -> i = j ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1367   F/wnf 1589   [wsb 1700   E*wmo 2253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257

This theorem is referenced by: (None)