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Theorem raaanlem 3528
Description: Special case of raaan 3529 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 2240 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
21cbvexv 1918 . . 3  |-  ( E. x  x  e.  A  <->  E. y  y  e.  A
)
3 raaan.1 . . . . 5  |-  F/ y
ph
43r19.28m 3512 . . . 4  |-  ( E. y  y  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  <->  ( ph  /\  A. y  e.  A  ps ) ) )
54ralbidv 2477 . . 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 121 . 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 2319 . . . 4  |-  F/_ x A
8 raaan.2 . . . 4  |-  F/ x ps
97, 8nfralxy 2515 . . 3  |-  F/ x A. y  e.  A  ps
109r19.27m 3518 . 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 188 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 104    <-> wb 105   F/wnf 1460   E.wex 1492    e. wcel 2148   A.wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460
This theorem is referenced by:  raaan  3529
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