Theorem ralf0 3344

Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)

Hypothesis

Ref	Expression
ralf0.1	|- -. ph

Assertion

Ref	Expression
ralf0	|- ( A. x e. A ph <-> A = (/) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem ralf0

Step	Hyp	Ref	Expression
1		ralf0.1	. . . . 5 |- -. ph
2		con3 603	. . . . 5 |- ( ( x e. A -> ph ) -> ( -. ph -> -. x e. A ) )
3	1, 2	mpi 15	. . . 4 |- ( ( x e. A -> ph ) -> -. x e. A )
4	3	alimi 1384	. . 3 |- ( A. x ( x e. A -> ph ) -> A. x -. x e. A )
5		df-ral 2353	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		eq0 3266	. . 3 |- ( A = (/) <-> A. x -. x e. A )
7	4, 5, 6	3imtr4i 199	. 2 |- ( A. x e. A ph -> A = (/) )
8		rzal 3338	. 2 |- ( A = (/) -> A. x e. A ph )
9	7, 8	impbii 124	1 |- ( A. x e. A ph <-> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   A.wral 2348   (/)c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-v 2603  df-dif 2975  df-nul 3252

This theorem is referenced by: (None)