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Theorem ceqsralt 2598
Description: Restricted quantifier version of ceqsalt 2597. (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 2328 . . . 4  |-  ( A. x  e.  B  (
x  =  A  ->  ph )  <->  A. x ( x  e.  B  ->  (
x  =  A  ->  ph ) ) )
2 eleq1 2116 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32pm5.32ri 436 . . . . . . . 8  |-  ( ( x  e.  B  /\  x  =  A )  <->  ( A  e.  B  /\  x  =  A )
)
43imbi1i 231 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =  A
)  ->  ph )  <->  ( ( A  e.  B  /\  x  =  A )  ->  ph ) )
5 impexp 254 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =  A
)  ->  ph )  <->  ( x  e.  B  ->  ( x  =  A  ->  ph )
) )
6 impexp 254 . . . . . . 7  |-  ( ( ( A  e.  B  /\  x  =  A
)  ->  ph )  <->  ( A  e.  B  ->  ( x  =  A  ->  ph )
) )
74, 5, 63bitr3i 203 . . . . . 6  |-  ( ( x  e.  B  -> 
( x  =  A  ->  ph ) )  <->  ( A  e.  B  ->  ( x  =  A  ->  ph )
) )
87albii 1375 . . . . 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 185 . . 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 1769 . . 3  |-  ( A. x ( A  e.  B  ->  ( x  =  A  ->  ph )
)  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) )
1210, 11syl6bb 189 . 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 234 . . 3  |-  ( A  e.  B  ->  ( A. x ( x  =  A  ->  ph )  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) ) )
14133ad2ant3 938 . 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 2597 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
1612, 14, 153bitr2d 209 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 101    <-> wb 102    /\ w3a 896   A.wal 1257    = wceq 1259   F/wnf 1365    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-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-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-v 2576
This theorem is referenced by:  ceqsralv  2602
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