MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ceqsralt Unicode version

Theorem ceqsralt 2824
Description: Restricted quantifier version of ceqsalt 2823. (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 2561 . . . 4  |-  ( A. x  e.  B  (
x  =  A  ->  ph )  <->  A. x ( x  e.  B  ->  (
x  =  A  ->  ph ) ) )
2 eleq1 2356 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32pm5.32ri 619 . . . . . . . 8  |-  ( ( x  e.  B  /\  x  =  A )  <->  ( A  e.  B  /\  x  =  A )
)
43imbi1i 315 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =  A
)  ->  ph )  <->  ( ( A  e.  B  /\  x  =  A )  ->  ph ) )
5 impexp 433 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =  A
)  ->  ph )  <->  ( x  e.  B  ->  ( x  =  A  ->  ph )
) )
6 impexp 433 . . . . . . 7  |-  ( ( ( A  e.  B  /\  x  =  A
)  ->  ph )  <->  ( A  e.  B  ->  ( x  =  A  ->  ph )
) )
74, 5, 63bitr3i 266 . . . . . 6  |-  ( ( x  e.  B  -> 
( x  =  A  ->  ph ) )  <->  ( A  e.  B  ->  ( x  =  A  ->  ph )
) )
87albii 1556 . . . . 5  |-  ( A. x ( x  e.  B  ->  ( x  =  A  ->  ph )
)  <->  A. x ( A  e.  B  ->  (
x  =  A  ->  ph ) ) )
98a1i 10 . . . 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 248 . . 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 1843 . . 3  |-  ( A. x ( A  e.  B  ->  ( x  =  A  ->  ph )
)  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) )
1210, 11syl6bb 252 . 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 325 . . 3  |-  ( A  e.  B  ->  ( A. x ( x  =  A  ->  ph )  <->  ( A  e.  B  ->  A. x
( x  =  A  ->  ph ) ) ) )
14133ad2ant3 978 . 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 2823 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
1612, 14, 153bitr2d 272 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    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   A.wral 2556
This theorem is referenced by:  ceqsralv  2828  cdleme32fva  31248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-v 2803
  Copyright terms: Public domain W3C validator