Theorem mo2icl 3245

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 2469	. . . . . 6 |- ( y = A -> ( x = y <-> x = A ) )
2	1	imbi2d 316	. . . . 5 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
3	2	albidv 1680	. . . 4 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
4	3	imbi1d 317	. . 3 |- ( y = A -> ( ( A. x ( ph -> x = y ) -> E* x ph ) <-> ( A. x ( ph -> x = A ) -> E* x ph ) ) )
5		19.8a 1797	. . . 4 |- ( A. x ( ph -> x = y ) -> E. y A. x ( ph -> x = y ) )
6		mo2v 2269	. . . 4 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
7	5, 6	sylibr 212	. . 3 |- ( A. x ( ph -> x = y ) -> E* x ph )
8	4, 7	vtoclg 3136	. 2 |- ( A e. _V -> ( A. x ( ph -> x = A ) -> E* x ph ) )
9		eqvisset 3086	. . . . . 6 |- ( x = A -> A e. _V )
10	9	imim2i 14	. . . . 5 |- ( ( ph -> x = A ) -> ( ph -> A e. _V ) )
11	10	con3rr3 136	. . . 4 |- ( -. A e. _V -> ( ( ph -> x = A ) -> -. ph ) )
12	11	alimdv 1676	. . 3 |- ( -. A e. _V -> ( A. x ( ph -> x = A ) -> A. x -. ph ) )
13		alnex 1589	. . . 4 |- ( A. x -. ph <-> -. E. x ph )
14		exmo 2291	. . . . 5 |- ( E. x ph \/ E* x ph )
15	14	ori 375	. . . 4 |- ( -. E. x ph -> E* x ph )
16	13, 15	sylbi 195	. . 3 |- ( A. x -. ph -> E* x ph )
17	12, 16	syl6 33	. 2 |- ( -. A e. _V -> ( A. x ( ph -> x = A ) -> E* x ph ) )
18	8, 17	pm2.61i 164	1 |- ( A. x ( ph -> x = A ) -> E* x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1368   = wceq 1370   E.wex 1587   e. wcel 1758   E*wmo 2263   _Vcvv 3078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3080

This theorem is referenced by:  invdisj  4392  opabiotafun  5864  fseqenlem2  8309  dfac2  8414  imasaddfnlem  14588  imasvscafn  14597  bnj149  32220