Theorem mo2icl 2982

Description: Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.)

Assertion

Ref	Expression
mo2icl	|- ( A. x ( ph -> x = A ) -> E* x ph )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem mo2icl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1587	. . . . 5 |- F/ x A. x ( ph -> x = A )
2		vex 2802	. . . . . . . 8 |- x e. _V
3		eleq1 2292	. . . . . . . 8 |- ( x = A -> ( x e. _V <-> A e. _V ) )
4	2, 3	mpbii 148	. . . . . . 7 |- ( x = A -> A e. _V )
5	4	imim2i 12	. . . . . 6 |- ( ( ph -> x = A ) -> ( ph -> A e. _V ) )
6	5	sps 1583	. . . . 5 |- ( A. x ( ph -> x = A ) -> ( ph -> A e. _V ) )
7	1, 6	eximd 1658	. . . 4 |- ( A. x ( ph -> x = A ) -> ( E. x ph -> E. x A e. _V ) )
8		19.9v 1917	. . . 4 |- ( E. x A e. _V <-> A e. _V )
9	7, 8	imbitrdi 161	. . 3 |- ( A. x ( ph -> x = A ) -> ( E. x ph -> A e. _V ) )
10		eqeq2 2239	. . . . . . . 8 |- ( y = A -> ( x = y <-> x = A ) )
11	10	imbi2d 230	. . . . . . 7 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
12	11	albidv 1870	. . . . . 6 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
13	12	imbi1d 231	. . . . 5 |- ( y = A -> ( ( A. x ( ph -> x = y ) -> E* x ph ) <-> ( A. x ( ph -> x = A ) -> E* x ph ) ) )
14		nfv 1574	. . . . . . 7 |- F/ y ph
15	14	mo2r 2130	. . . . . 6 |- ( E. y A. x ( ph -> x = y ) -> E* x ph )
16	15	19.23bi 1638	. . . . 5 |- ( A. x ( ph -> x = y ) -> E* x ph )
17	13, 16	vtoclg 2861	. . . 4 |- ( A e. _V -> ( A. x ( ph -> x = A ) -> E* x ph ) )
18	17	com12 30	. . 3 |- ( A. x ( ph -> x = A ) -> ( A e. _V -> E* x ph ) )
19	9, 18	syld 45	. 2 |- ( A. x ( ph -> x = A ) -> ( E. x ph -> E* x ph ) )
20		moabs 2127	. 2 |- ( E* x ph <-> ( E. x ph -> E* x ph ) )
21	19, 20	sylibr 134	1 |- ( A. x ( ph -> x = A ) -> E* x ph )

Syntax hints:   -> wi 4   A.wal 1393   = wceq 1395   E.wex 1538   E*wmo 2078   e. wcel 2200   _Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  invdisj  4075  imasaddfnlemg  13342