Theorem nmo 29325

Description: Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)

Hypothesis

Ref	Expression
nmo.1	|- F/ y ph

Assertion

Ref	Expression
nmo	|- ( -. E* x ph <-> A. y E. x ( ph /\ x =/= y ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem nmo

Step	Hyp	Ref	Expression
1		nmo.1	. . . 4 |- F/ y ph
2	1	mo2 2479	. . 3 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
3	2	notbii 310	. 2 |- ( -. E* x ph <-> -. E. y A. x ( ph -> x = y ) )
4		alnex 1706	. 2 |- ( A. y -. A. x ( ph -> x = y ) <-> -. E. y A. x ( ph -> x = y ) )
5		exnal 1754	. . . 4 |- ( E. x -. ( ph -> x = y ) <-> -. A. x ( ph -> x = y ) )
6		pm4.61 442	. . . . . 6 |- ( -. ( ph -> x = y ) <-> ( ph /\ -. x = y ) )
7		biid 251	. . . . . . . 8 |- ( x = y <-> x = y )
8	7	necon3bbii 2841	. . . . . . 7 |- ( -. x = y <-> x =/= y )
9	8	anbi2i 730	. . . . . 6 |- ( ( ph /\ -. x = y ) <-> ( ph /\ x =/= y ) )
10	6, 9	bitri 264	. . . . 5 |- ( -. ( ph -> x = y ) <-> ( ph /\ x =/= y ) )
11	10	exbii 1774	. . . 4 |- ( E. x -. ( ph -> x = y ) <-> E. x ( ph /\ x =/= y ) )
12	5, 11	bitr3i 266	. . 3 |- ( -. A. x ( ph -> x = y ) <-> E. x ( ph /\ x =/= y ) )
13	12	albii 1747	. 2 |- ( A. y -. A. x ( ph -> x = y ) <-> A. y E. x ( ph /\ x =/= y ) )
14	3, 4, 13	3bitr2i 288	1 |- ( -. E* x ph <-> A. y E. x ( ph /\ x =/= y ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   E*wmo 2471   =/= wne 2794

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-ne 2795

This theorem is referenced by: (None)