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Theorem rspn0 3744
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 3741 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 nfra1 2872 . . . 4  |-  F/ x A. x  e.  A  ph
3 nfv 1674 . . . 4  |-  F/ x ph
42, 3nfim 1855 . . 3  |-  F/ x
( A. x  e.  A  ph  ->  ph )
5 rsp 2881 . . . 4  |-  ( A. x  e.  A  ph  ->  ( x  e.  A  ->  ph ) )
65com12 31 . . 3  |-  ( x  e.  A  ->  ( A. x  e.  A  ph 
->  ph ) )
74, 6exlimi 1847 . 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 1587    e. wcel 1758    =/= wne 2642   A.wral 2793   (/)c0 3732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-v 3067  df-dif 3426  df-nul 3733
This theorem is referenced by:  hashge2el2dif  12283  ralralimp  30254
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