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Theorem ceqsralt 2708
Description: Restricted quantifier version of ceqsalt 2707. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
Assertion
Ref Expression
ceqsralt  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem ceqsralt
StepHypRef Expression
1 df-ral 2419 . . . 4  |-  ( A. x  e.  B  (
x  =  A  ->  ph )  <->  A. x ( x  e.  B  ->  (
x  =  A  ->  ph ) ) )
2 eleq1 2200 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32pm5.32ri 450 . . . . . . . 8  |-  ( ( x  e.  B  /\  x  =  A )  <->  ( A  e.  B  /\  x  =  A )
)
43imbi1i 237 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =  A
)  ->  ph )  <->  ( ( A  e.  B  /\  x  =  A )  ->  ph ) )
5 impexp 261 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =  A
)  ->  ph )  <->  ( x  e.  B  ->  ( x  =  A  ->  ph )
) )
6 impexp 261 . . . . . . 7  |-  ( ( ( A  e.  B  /\  x  =  A
)  ->  ph )  <->  ( A  e.  B  ->  ( x  =  A  ->  ph )
) )
74, 5, 63bitr3i 209 . . . . . 6  |-  ( ( x  e.  B  -> 
( x  =  A  ->  ph ) )  <->  ( A  e.  B  ->  ( x  =  A  ->  ph )
) )
87albii 1446 . . . . 5  |-  ( A. x ( x  e.  B  ->  ( x  =  A  ->  ph )
)  <->  A. x ( A  e.  B  ->  (
x  =  A  ->  ph ) ) )
98a1i 9 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x ( x  e.  B  ->  ( x  =  A  ->  ph )
)  <->  A. x ( A  e.  B  ->  (
x  =  A  ->  ph ) ) ) )
101, 9syl5bb 191 . . 3  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  A. x
( A  e.  B  ->  ( x  =  A  ->  ph ) ) ) )
11 19.21v 1845 . . 3  |-  ( A. x ( A  e.  B  ->  ( x  =  A  ->  ph )
)  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) )
1210, 11syl6bb 195 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) ) )
13 biimt 240 . . 3  |-  ( A  e.  B  ->  ( A. x ( x  =  A  ->  ph )  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) ) )
14133ad2ant3 1004 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x ( x  =  A  ->  ph )  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) ) )
15 ceqsalt 2707 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
1612, 14, 153bitr2d 215 1  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962   A.wal 1329    = wceq 1331   F/wnf 1436    e. wcel 1480   A.wral 2414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-v 2683
This theorem is referenced by:  ceqsralv  2712
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