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Theorem ceqsrexv 3082
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 2711 . . 3  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  E. x ( x  e.  B  /\  (
x  =  A  /\  ph ) ) )
2 an12 788 . . . 4  |-  ( ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  ( x  e.  B  /\  (
x  =  A  /\  ph ) ) )
32exbii 1634 . . 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 2493 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 ceqsrexv.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6anbi12d 703 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
87ceqsexgv 3081 . . 3  |-  ( A  e.  B  ->  ( E. x ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  ( A  e.  B  /\  ps )
) )
98bianabs 868 . 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 369    = wceq 1362   E.wex 1589    e. wcel 1755   E.wrex 2706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-rex 2711  df-v 2964
This theorem is referenced by:  ceqsrexbv  3083  ceqsrex2v  3084  reuxfr2d  4503  f1oiso  6029  creur  10304  creui  10305  deg1ldg  21448  ulm2  21735  reuxfr3d  25697  rmxdiophlem  29209  expdiophlem1  29215  expdiophlem2  29216  eqlkr3  32319  diclspsn  34412
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