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Theorem raaanlem 3354
Description: Special case of raaan 3355 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 2116 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
21cbvexv 1811 . . 3  |-  ( E. x  x  e.  A  <->  E. y  y  e.  A
)
3 raaan.1 . . . . 5  |-  F/ y
ph
43r19.28m 3339 . . . 4  |-  ( E. y  y  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  <->  ( ph  /\  A. y  e.  A  ps ) ) )
54ralbidv 2343 . . 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 118 . 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 2194 . . . 4  |-  F/_ x A
8 raaan.2 . . . 4  |-  F/ x ps
97, 8nfralxy 2377 . . 3  |-  F/ x A. y  e.  A  ps
109r19.27m 3344 . 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 181 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 101    <-> wb 102   F/wnf 1365   E.wex 1397    e. wcel 1409   A.wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328
This theorem is referenced by:  raaan  3355
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