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Theorem ceqsrexv 2914
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 2562 . . 3  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  E. x ( x  e.  B  /\  (
x  =  A  /\  ph ) ) )
2 an12 772 . . . 4  |-  ( ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  ( x  e.  B  /\  (
x  =  A  /\  ph ) ) )
32exbii 1572 . . 3  |-  ( E. x ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  E. x
( x  e.  B  /\  ( x  =  A  /\  ph ) ) )
41, 3bitr4i 243 . 2  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  E. x ( x  =  A  /\  (
x  e.  B  /\  ph ) ) )
5 eleq1 2356 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 ceqsrexv.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6anbi12d 691 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
87ceqsexgv 2913 . . 3  |-  ( A  e.  B  ->  ( E. x ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  ( A  e.  B  /\  ps )
) )
98bianabs 850 . 2  |-  ( A  e.  B  ->  ( E. x ( x  =  A  /\  ( x  e.  B  /\  ph ) )  <->  ps )
)
104, 9syl5bb 248 1  |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557
This theorem is referenced by:  ceqsrexbv  2915  ceqsrex2v  2916  reuxfr2d  4573  f1oiso  5864  creur  9756  creui  9757  deg1ldg  19494  ulm2  19780  reuxfr3d  23154  ceqsrexv2  24093  rmxdiophlem  27211  expdiophlem1  27217  expdiophlem2  27218  eqlkr3  29913  diclspsn  32006
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-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803
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