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Theorem ralf0 3783
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
StepHypRef Expression
1 ralf0.1 . . . . 5  |-  -.  ph
2 con3 134 . . . . 5  |-  ( ( x  e.  A  ->  ph )  ->  ( -. 
ph  ->  -.  x  e.  A ) )
31, 2mpi 17 . . . 4  |-  ( ( x  e.  A  ->  ph )  ->  -.  x  e.  A )
43alimi 1609 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  ->  A. x  -.  x  e.  A )
5 df-ral 2718 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
6 eq0 3649 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
74, 5, 63imtr4i 266 . 2  |-  ( A. x  e.  A  ph  ->  A  =  (/) )
8 rzal 3778 . 2  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
97, 8impbii 188 1  |-  ( A. x  e.  A  ph  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1362    = wceq 1364    e. wcel 1761   A.wral 2713   (/)c0 3634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-v 2972  df-dif 3328  df-nul 3635
This theorem is referenced by:  uvtx01vtx  23335  rusgra0edg  30498
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