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Theorem ralf0 3698
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 128 . . . . 5  |-  ( ( x  e.  A  ->  ph )  ->  ( -. 
ph  ->  -.  x  e.  A ) )
31, 2mpi 17 . . . 4  |-  ( ( x  e.  A  ->  ph )  ->  -.  x  e.  A )
43alimi 1565 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  ->  A. x  -.  x  e.  A )
5 df-ral 2675 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
6 eq0 3606 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
74, 5, 63imtr4i 258 . 2  |-  ( A. x  e.  A  ph  ->  A  =  (/) )
8 rzal 3693 . 2  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
97, 8impbii 181 1  |-  ( A. x  e.  A  ph  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1721   A.wral 2670   (/)c0 3592
This theorem is referenced by:  uvtx01vtx  21458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-v 2922  df-dif 3287  df-nul 3593
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