Theorem ralf0 3518

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 637	. . . . 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 1448	. . 3 |- ( A. x ( x e. A -> ph ) -> A. x -. x e. A )
5		df-ral 2453	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		eq0 3433	. . 3 |- ( A = (/) <-> A. x -. x e. A )
7	4, 5, 6	3imtr4i 200	. 2 |- ( A. x e. A ph -> A = (/) )
8		rzal 3512	. 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 1346   = wceq 1348   e. wcel 2141   A.wral 2448   (/)c0 3414

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 609  ax-in2 610  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-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by: (None)