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Theorem ralnralall 32128
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 2968 . 2  |-  ( A. x  e.  A  ( ph  /\  -.  ph )  <->  ( A. x  e.  A  ph 
/\  A. x  e.  A  -.  ph ) )
2 pm3.24 880 . . . . 5  |-  -.  ( ph  /\  -.  ph )
32bifal 1394 . . . 4  |-  ( (
ph  /\  -.  ph )  <-> F.  )
43ralbii 2872 . . 3  |-  ( A. x  e.  A  ( ph  /\  -.  ph )  <->  A. x  e.  A F.  )
5 r19.3rzv 3904 . . . 4  |-  ( A  =/=  (/)  ->  ( F.  <->  A. x  e.  A F.  ) )
6 falim 1395 . . . 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 369   F. wfal 1386    =/= wne 2636   A.wral 2791   (/)c0 3767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-v 3095  df-dif 3461  df-nul 3768
This theorem is referenced by: (None)
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