Theorem mo2icl 2909

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 1534	. . . . 5 |- F/ x A. x ( ph -> x = A )
2		vex 2733	. . . . . . . 8 |- x e. _V
3		eleq1 2233	. . . . . . . 8 |- ( x = A -> ( x e. _V <-> A e. _V ) )
4	2, 3	mpbii 147	. . . . . . 7 |- ( x = A -> A e. _V )
5	4	imim2i 12	. . . . . 6 |- ( ( ph -> x = A ) -> ( ph -> A e. _V ) )
6	5	sps 1530	. . . . 5 |- ( A. x ( ph -> x = A ) -> ( ph -> A e. _V ) )
7	1, 6	eximd 1605	. . . 4 |- ( A. x ( ph -> x = A ) -> ( E. x ph -> E. x A e. _V ) )
8		19.9v 1864	. . . 4 |- ( E. x A e. _V <-> A e. _V )
9	7, 8	syl6ib 160	. . 3 |- ( A. x ( ph -> x = A ) -> ( E. x ph -> A e. _V ) )
10		eqeq2 2180	. . . . . . . 8 |- ( y = A -> ( x = y <-> x = A ) )
11	10	imbi2d 229	. . . . . . 7 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
12	11	albidv 1817	. . . . . 6 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
13	12	imbi1d 230	. . . . 5 |- ( y = A -> ( ( A. x ( ph -> x = y ) -> E* x ph ) <-> ( A. x ( ph -> x = A ) -> E* x ph ) ) )
14		nfv 1521	. . . . . . 7 |- F/ y ph
15	14	mo2r 2071	. . . . . 6 |- ( E. y A. x ( ph -> x = y ) -> E* x ph )
16	15	19.23bi 1585	. . . . 5 |- ( A. x ( ph -> x = y ) -> E* x ph )
17	13, 16	vtoclg 2790	. . . 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 2068	. 2 |- ( E* x ph <-> ( E. x ph -> E* x ph ) )
21	19, 20	sylibr 133	1 |- ( A. x ( ph -> x = A ) -> E* x ph )

Syntax hints:   -> wi 4   A.wal 1346   = wceq 1348   E.wex 1485   E*wmo 2020   e. wcel 2141   _Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  invdisj  3983