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Theorem rspn0 3796
Description: Specialization for restricted generalization with a nonempty set. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
Assertion
Ref Expression
rspn0  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ph  ->  ph ) )
Distinct variable groups:    x, A    ph, x

Proof of Theorem rspn0
StepHypRef Expression
1 n0 3793 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 nfra1 2835 . . . 4  |-  F/ x A. x  e.  A  ph
3 nfv 1712 . . . 4  |-  F/ x ph
42, 3nfim 1925 . . 3  |-  F/ x
( A. x  e.  A  ph  ->  ph )
5 rsp 2820 . . . 4  |-  ( A. x  e.  A  ph  ->  ( x  e.  A  ->  ph ) )
65com12 31 . . 3  |-  ( x  e.  A  ->  ( A. x  e.  A  ph 
->  ph ) )
74, 6exlimi 1917 . 2  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ph  ->  ph )
)
81, 7sylbi 195 1  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ph  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1617    e. wcel 1823    =/= wne 2649   A.wral 2804   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-v 3108  df-dif 3464  df-nul 3784
This theorem is referenced by:  hashge2el2dif  12508  scmatf1  19203  usgfiregdegfi  25116  ralralimp  32688
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