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Theorem raaanlem 3326
Description: Special case of raaan 3327 where  A is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
Hypotheses
Ref Expression
raaan.1  |-  F/ y
ph
raaan.2  |-  F/ x ps
Assertion
Ref Expression
raaanlem  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) ) )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem raaanlem
StepHypRef Expression
1 eleq1 2100 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
21cbvexv 1795 . . 3  |-  ( E. x  x  e.  A  <->  E. y  y  e.  A
)
3 raaan.1 . . . . 5  |-  F/ y
ph
43r19.28m 3311 . . . 4  |-  ( E. y  y  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  <->  ( ph  /\  A. y  e.  A  ps ) ) )
54ralbidv 2326 . . 3  |-  ( E. y  y  e.  A  ->  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  A. x  e.  A  ( ph  /\  A. y  e.  A  ps )
) )
62, 5sylbi 114 . 2  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  A. x  e.  A  ( ph  /\  A. y  e.  A  ps )
) )
7 nfcv 2178 . . . 4  |-  F/_ x A
8 raaan.2 . . . 4  |-  F/ x ps
97, 8nfralxy 2360 . . 3  |-  F/ x A. y  e.  A  ps
109r19.27m 3316 . 2  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  A. y  e.  A  ps ) 
<->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) ) )
116, 10bitrd 177 1  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98   F/wnf 1349   E.wex 1381    e. wcel 1393   A.wral 2306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311
This theorem is referenced by:  raaan  3327
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