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Theorem ceqsralt 3058
Description: Restricted quantifier version of ceqsalt 3057. (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 2737 . . . 4  |-  ( A. x  e.  B  (
x  =  A  ->  ph )  <->  A. x ( x  e.  B  ->  (
x  =  A  ->  ph ) ) )
2 eleq1 2454 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32pm5.32ri 636 . . . . . . . 8  |-  ( ( x  e.  B  /\  x  =  A )  <->  ( A  e.  B  /\  x  =  A )
)
43imbi1i 323 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =  A
)  ->  ph )  <->  ( ( A  e.  B  /\  x  =  A )  ->  ph ) )
5 impexp 444 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =  A
)  ->  ph )  <->  ( x  e.  B  ->  ( x  =  A  ->  ph )
) )
6 impexp 444 . . . . . . 7  |-  ( ( ( A  e.  B  /\  x  =  A
)  ->  ph )  <->  ( A  e.  B  ->  ( x  =  A  ->  ph )
) )
74, 5, 63bitr3i 275 . . . . . 6  |-  ( ( x  e.  B  -> 
( x  =  A  ->  ph ) )  <->  ( A  e.  B  ->  ( x  =  A  ->  ph )
) )
87albii 1648 . . . . 5  |-  ( A. x ( x  e.  B  ->  ( x  =  A  ->  ph )
)  <->  A. x ( A  e.  B  ->  (
x  =  A  ->  ph ) ) )
98a1i 11 . . . 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 257 . . 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 1737 . . 3  |-  ( A. x ( A  e.  B  ->  ( x  =  A  ->  ph )
)  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) )
1210, 11syl6bb 261 . 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 333 . . 3  |-  ( A  e.  B  ->  ( A. x ( x  =  A  ->  ph )  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) ) )
14133ad2ant3 1017 . 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 3057 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
1612, 14, 153bitr2d 281 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   A.wal 1397    = wceq 1399   F/wnf 1624    e. wcel 1826   A.wral 2732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-12 1862  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-ral 2737  df-v 3036
This theorem is referenced by:  ceqsralv  3063  cdleme32fva  36576
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