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Theorem ceqsrexv 3158
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
Hypothesis
Ref Expression
ceqsrexv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsrexv  |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ps )
)
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsrexv
StepHypRef Expression
1 df-rex 2738 . . 3  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  E. x ( x  e.  B  /\  (
x  =  A  /\  ph ) ) )
2 an12 795 . . . 4  |-  ( ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  ( x  e.  B  /\  (
x  =  A  /\  ph ) ) )
32exbii 1675 . . 3  |-  ( E. x ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  E. x
( x  e.  B  /\  ( x  =  A  /\  ph ) ) )
41, 3bitr4i 252 . 2  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  E. x ( x  =  A  /\  (
x  e.  B  /\  ph ) ) )
5 eleq1 2454 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 ceqsrexv.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6anbi12d 708 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
87ceqsexgv 3157 . . 3  |-  ( A  e.  B  ->  ( E. x ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  ( A  e.  B  /\  ps )
) )
98bianabs 878 . 2  |-  ( A  e.  B  ->  ( E. x ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  ps )
)
104, 9syl5bb 257 1  |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399   E.wex 1620    e. wcel 1826   E.wrex 2733
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-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-rex 2738  df-v 3036
This theorem is referenced by:  ceqsrexbv  3159  ceqsrex2v  3160  reuxfr2d  4585  f1oiso  6148  creur  10446  creui  10447  deg1ldg  22577  ulm2  22865  reuxfr3d  27505  rmxdiophlem  31123  expdiophlem1  31129  expdiophlem2  31130  eqlkr3  35239  diclspsn  37334
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