Theorem mo2icl 2720

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 1434	. . . . 5 |- F/ x A. x ( ph -> x = A )
2		vex 2560	. . . . . . . 8 |- x e. _V
3		eleq1 2100	. . . . . . . 8 |- ( x = A -> ( x e. _V <-> A e. _V ) )
4	2, 3	mpbii 136	. . . . . . 7 |- ( x = A -> A e. _V )
5	4	imim2i 12	. . . . . 6 |- ( ( ph -> x = A ) -> ( ph -> A e. _V ) )
6	5	sps 1430	. . . . 5 |- ( A. x ( ph -> x = A ) -> ( ph -> A e. _V ) )
7	1, 6	eximd 1503	. . . 4 |- ( A. x ( ph -> x = A ) -> ( E. x ph -> E. x A e. _V ) )
8		19.9v 1751	. . . 4 |- ( E. x A e. _V <-> A e. _V )
9	7, 8	syl6ib 150	. . 3 |- ( A. x ( ph -> x = A ) -> ( E. x ph -> A e. _V ) )
10		eqeq2 2049	. . . . . . . 8 |- ( y = A -> ( x = y <-> x = A ) )
11	10	imbi2d 219	. . . . . . 7 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
12	11	albidv 1705	. . . . . 6 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
13	12	imbi1d 220	. . . . 5 |- ( y = A -> ( ( A. x ( ph -> x = y ) -> E* x ph ) <-> ( A. x ( ph -> x = A ) -> E* x ph ) ) )
14		nfv 1421	. . . . . . 7 |- F/ y ph
15	14	mo2r 1952	. . . . . 6 |- ( E. y A. x ( ph -> x = y ) -> E* x ph )
16	15	19.23bi 1483	. . . . 5 |- ( A. x ( ph -> x = y ) -> E* x ph )
17	13, 16	vtoclg 2613	. . . 4 |- ( A e. _V -> ( A. x ( ph -> x = A ) -> E* x ph ) )
18	17	com12 27	. . 3 |- ( A. x ( ph -> x = A ) -> ( A e. _V -> E* x ph ) )
19	9, 18	syld 40	. 2 |- ( A. x ( ph -> x = A ) -> ( E. x ph -> E* x ph ) )
20		moabs 1949	. 2 |- ( E* x ph <-> ( E. x ph -> E* x ph ) )
21	19, 20	sylibr 137	1 |- ( A. x ( ph -> x = A ) -> E* x ph )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   E*wmo 1901   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  invdisj  3759