ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralf0 Unicode version

Theorem ralf0 3381
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 606 . . . . 5  |-  ( ( x  e.  A  ->  ph )  ->  ( -. 
ph  ->  -.  x  e.  A ) )
31, 2mpi 15 . . . 4  |-  ( ( x  e.  A  ->  ph )  ->  -.  x  e.  A )
43alimi 1389 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  ->  A. x  -.  x  e.  A )
5 df-ral 2364 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
6 eq0 3299 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
74, 5, 63imtr4i 199 . 2  |-  ( A. x  e.  A  ph  ->  A  =  (/) )
8 rzal 3375 . 2  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
97, 8impbii 124 1  |-  ( A. x  e.  A  ph  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   A.wral 2359   (/)c0 3284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 2999  df-nul 3285
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator