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Theorem ralnralall 37927
Description: A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
Assertion
Ref Expression
ralnralall  |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
/\  A. x  e.  A  -.  ph )  ->  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem ralnralall
StepHypRef Expression
1 r19.26 2934 . 2  |-  ( A. x  e.  A  ( ph  /\  -.  ph )  <->  ( A. x  e.  A  ph 
/\  A. x  e.  A  -.  ph ) )
2 pm3.24 883 . . . . 5  |-  -.  ( ph  /\  -.  ph )
32bifal 1418 . . . 4  |-  ( (
ph  /\  -.  ph )  <-> F.  )
43ralbii 2835 . . 3  |-  ( A. x  e.  A  ( ph  /\  -.  ph )  <->  A. x  e.  A F.  )
5 r19.3rzv 3866 . . . 4  |-  ( A  =/=  (/)  ->  ( F.  <->  A. x  e.  A F.  ) )
6 falim 1419 . . . 4  |-  ( F. 
->  ps )
75, 6syl6bir 229 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A F.  ->  ps ) )
84, 7syl5bi 217 . 2  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  -.  ph )  ->  ps ) )
91, 8syl5bir 218 1  |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
/\  A. x  e.  A  -.  ph )  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   F. wfal 1410    =/= wne 2598   A.wral 2754   (/)c0 3738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-v 3061  df-dif 3417  df-nul 3739
This theorem is referenced by: (None)
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