Theorem mo2icl 3228

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 2472	. . . . . 6 |- ( y = A -> ( x = y <-> x = A ) )
2	1	imbi2d 322	. . . . 5 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
3	2	albidv 1777	. . . 4 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
4	3	imbi1d 323	. . 3 |- ( y = A -> ( ( A. x ( ph -> x = y ) -> E* x ph ) <-> ( A. x ( ph -> x = A ) -> E* x ph ) ) )
5		19.8a 1945	. . . 4 |- ( A. x ( ph -> x = y ) -> E. y A. x ( ph -> x = y ) )
6		mo2v 2316	. . . 4 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
7	5, 6	sylibr 217	. . 3 |- ( A. x ( ph -> x = y ) -> E* x ph )
8	4, 7	vtoclg 3118	. 2 |- ( A e. _V -> ( A. x ( ph -> x = A ) -> E* x ph ) )
9		eqvisset 3064	. . . . . 6 |- ( x = A -> A e. _V )
10	9	imim2i 16	. . . . 5 |- ( ( ph -> x = A ) -> ( ph -> A e. _V ) )
11	10	con3rr3 143	. . . 4 |- ( -. A e. _V -> ( ( ph -> x = A ) -> -. ph ) )
12	11	alimdv 1773	. . 3 |- ( -. A e. _V -> ( A. x ( ph -> x = A ) -> A. x -. ph ) )
13		alnex 1675	. . . 4 |- ( A. x -. ph <-> -. E. x ph )
14		exmo 2334	. . . . 5 |- ( E. x ph \/ E* x ph )
15	14	ori 381	. . . 4 |- ( -. E. x ph -> E* x ph )
16	13, 15	sylbi 200	. . 3 |- ( A. x -. ph -> E* x ph )
17	12, 16	syl6 34	. 2 |- ( -. A e. _V -> ( A. x ( ph -> x = A ) -> E* x ph ) )
18	8, 17	pm2.61i 169	1 |- ( A. x ( ph -> x = A ) -> E* x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1452   = wceq 1454   E.wex 1673   e. wcel 1897   E*wmo 2310   _Vcvv 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3058

This theorem is referenced by:  invdisj  4404  opabiotafun  5948  fseqenlem2  8481  dfac2  8586  imasaddfnlem  15482  imasvscafn  15491  bnj149  29734