Theorem ralf0 3597

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 647	. . . . 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 1503	. . 3 |- ( A. x ( x e. A -> ph ) -> A. x -. x e. A )
5		df-ral 2515	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		eq0 3513	. . 3 |- ( A = (/) <-> A. x -. x e. A )
7	4, 5, 6	3imtr4i 201	. 2 |- ( A. x e. A ph -> A = (/) )
8		rzal 3592	. 2 |- ( A = (/) -> A. x e. A ph )
9	7, 8	impbii 126	1 |- ( A. x e. A ph <-> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   A.wal 1395   = wceq 1397   e. wcel 2202   A.wral 2510   (/)c0 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-v 2804  df-dif 3202  df-nul 3495

This theorem is referenced by: (None)