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Theorem ralf0 3560
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 126 . . . . 5  |-  ( ( x  e.  A  ->  ph )  ->  ( -. 
ph  ->  -.  x  e.  A ) )
31, 2mpi 16 . . . 4  |-  ( ( x  e.  A  ->  ph )  ->  -.  x  e.  A )
43alimi 1546 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  ->  A. x  -.  x  e.  A )
5 df-ral 2548 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
6 eq0 3469 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
74, 5, 63imtr4i 257 . 2  |-  ( A. x  e.  A  ph  ->  A  =  (/) )
8 rzal 3555 . 2  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
97, 8impbii 180 1  |-  ( A. x  e.  A  ph  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543   (/)c0 3455
This theorem is referenced by:  uvtx01vtx  28164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-nul 3456
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