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Theorem ceqsrexbv 2902
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
Hypothesis
Ref Expression
ceqsrexv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsrexbv  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ( A  e.  B  /\  ps )
)
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsrexbv
StepHypRef Expression
1 r19.42v 2694 . 2  |-  ( E. x  e.  B  ( A  e.  B  /\  ( x  =  A  /\  ph ) )  <->  ( A  e.  B  /\  E. x  e.  B  ( x  =  A  /\  ph )
) )
2 eleq1 2343 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32adantr 451 . . . . . 6  |-  ( ( x  =  A  /\  ph )  ->  ( x  e.  B  <->  A  e.  B
) )
43pm5.32ri 619 . . . . 5  |-  ( ( x  e.  B  /\  ( x  =  A  /\  ph ) )  <->  ( A  e.  B  /\  (
x  =  A  /\  ph ) ) )
54bicomi 193 . . . 4  |-  ( ( A  e.  B  /\  ( x  =  A  /\  ph ) )  <->  ( x  e.  B  /\  (
x  =  A  /\  ph ) ) )
65baib 871 . . 3  |-  ( x  e.  B  ->  (
( A  e.  B  /\  ( x  =  A  /\  ph ) )  <-> 
( x  =  A  /\  ph ) ) )
76rexbiia 2576 . 2  |-  ( E. x  e.  B  ( A  e.  B  /\  ( x  =  A  /\  ph ) )  <->  E. x  e.  B  ( x  =  A  /\  ph )
)
8 ceqsrexv.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
98ceqsrexv 2901 . . 3  |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ps )
)
109pm5.32i 618 . 2  |-  ( ( A  e.  B  /\  E. x  e.  B  ( x  =  A  /\  ph ) )  <->  ( A  e.  B  /\  ps )
)
111, 7, 103bitr3i 266 1  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ( A  e.  B  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544
This theorem is referenced by:  marypha2lem2  7189  txkgen  17346  eq0rabdioph  26856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790
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