Theorem mo2icl 3385

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		eqeq2 2633	. . . . . 6 |- ( y = A -> ( x = y <-> x = A ) )
2	1	imbi2d 330	. . . . 5 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
3	2	albidv 1849	. . . 4 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
4	3	imbi1d 331	. . 3 |- ( y = A -> ( ( A. x ( ph -> x = y ) -> E* x ph ) <-> ( A. x ( ph -> x = A ) -> E* x ph ) ) )
5		19.8a 2052	. . . 4 |- ( A. x ( ph -> x = y ) -> E. y A. x ( ph -> x = y ) )
6		mo2v 2477	. . . 4 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
7	5, 6	sylibr 224	. . 3 |- ( A. x ( ph -> x = y ) -> E* x ph )
8	4, 7	vtoclg 3266	. 2 |- ( A e. _V -> ( A. x ( ph -> x = A ) -> E* x ph ) )
9		eqvisset 3211	. . . . . 6 |- ( x = A -> A e. _V )
10	9	imim2i 16	. . . . 5 |- ( ( ph -> x = A ) -> ( ph -> A e. _V ) )
11	10	con3rr3 151	. . . 4 |- ( -. A e. _V -> ( ( ph -> x = A ) -> -. ph ) )
12	11	alimdv 1845	. . 3 |- ( -. A e. _V -> ( A. x ( ph -> x = A ) -> A. x -. ph ) )
13		alnex 1706	. . . 4 |- ( A. x -. ph <-> -. E. x ph )
14		exmo 2495	. . . . 5 |- ( E. x ph \/ E* x ph )
15	14	ori 390	. . . 4 |- ( -. E. x ph -> E* x ph )
16	13, 15	sylbi 207	. . 3 |- ( A. x -. ph -> E* x ph )
17	12, 16	syl6 35	. 2 |- ( -. A e. _V -> ( A. x ( ph -> x = A ) -> E* x ph ) )
18	8, 17	pm2.61i 176	1 |- ( A. x ( ph -> x = A ) -> E* x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E*wmo 2471   _Vcvv 3200

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-9 1999  ax-10 2019  ax-12 2047  ax-ext 2602

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  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  invdisj  4638  reusv1  4866  reusv2lem1  4868  opabiotafun  6259  fseqenlem2  8848  dfac2  8953  imasaddfnlem  16188  imasvscafn  16197  bnj149  30945