Theorem ralf0 3657

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 126	. . . . 5 |- ((x e. A -> ph) -> (-. ph -> -. x e. A))
3	1, 2	mpi 16	. . . 4 |- ((x e. A -> ph) -> -. x e. A)
4	3	alimi 1559	. . 3 |- (A.x(x e. A -> ph) -> A.x -. x e. A)
5		df-ral 2620	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
6		eq0 3565	. . 3 |- (A = (/) <-> A.x -. x e. A)
7	4, 5, 6	3imtr4i 257	. 2 |- (A.x e. A ph -> A = (/))
8		rzal 3652	. 2 |- (A = (/) -> A.x e. A ph)
9	7, 8	impbii 180	1 |- (A.x e. A ph <-> A = (/))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  A.wral 2615  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)