Theorem ralf0 3889

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 134	. . . . 5 |- ( ( x e. A -> ph ) -> ( -. ph -> -. x e. A ) )
3	1, 2	mpi 17	. . . 4 |- ( ( x e. A -> ph ) -> -. x e. A )
4	3	alimi 1605	. . 3 |- ( A. x ( x e. A -> ph ) -> A. x -. x e. A )
5		df-ral 2801	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		eq0 3755	. . 3 |- ( A = (/) <-> A. x -. x e. A )
7	4, 5, 6	3imtr4i 266	. 2 |- ( A. x e. A ph -> A = (/) )
8		rzal 3884	. 2 |- ( A = (/) -> A. x e. A ph )
9	7, 8	impbii 188	1 |- ( A. x e. A ph <-> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   A.wal 1368   = wceq 1370   e. wcel 1758   A.wral 2796   (/)c0 3740

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-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431

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-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-v 3074  df-dif 3434  df-nul 3741

This theorem is referenced by:  uvtx01vtx  23547  rusgra0edg  30716