Theorem ralf0 3497

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 632	. . . . 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 1435	. . 3 |- ( A. x ( x e. A -> ph ) -> A. x -. x e. A )
5		df-ral 2440	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		eq0 3412	. . 3 |- ( A = (/) <-> A. x -. x e. A )
7	4, 5, 6	3imtr4i 200	. 2 |- ( A. x e. A ph -> A = (/) )
8		rzal 3491	. 2 |- ( A = (/) -> A. x e. A ph )
9	7, 8	impbii 125	1 |- ( A. x e. A ph <-> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   A.wal 1333   = wceq 1335   e. wcel 2128   A.wral 2435   (/)c0 3394

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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-v 2714  df-dif 3104  df-nul 3395

This theorem is referenced by: (None)