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Theorem ralf0 3434
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 614 . . . . 5  |-  ( ( x  e.  A  ->  ph )  ->  ( -. 
ph  ->  -.  x  e.  A ) )
31, 2mpi 15 . . . 4  |-  ( ( x  e.  A  ->  ph )  ->  -.  x  e.  A )
43alimi 1414 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  ->  A. x  -.  x  e.  A )
5 df-ral 2396 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
6 eq0 3349 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
74, 5, 63imtr4i 200 . 2  |-  ( A. x  e.  A  ph  ->  A  =  (/) )
8 rzal 3428 . 2  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
97, 8impbii 125 1  |-  ( A. x  e.  A  ph  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104   A.wal 1312    = wceq 1314    e. wcel 1463   A.wral 2391   (/)c0 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-v 2660  df-dif 3041  df-nul 3332
This theorem is referenced by: (None)
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