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Theorem ceqsrexbv 3148
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 2922 . 2  |-  ( E. x  e.  B  ( A  e.  B  /\  ( x  =  A  /\  ph ) )  <->  ( A  e.  B  /\  E. x  e.  B  ( x  =  A  /\  ph )
) )
2 eleq1 2494 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32adantr 466 . . . . . 6  |-  ( ( x  =  A  /\  ph )  ->  ( x  e.  B  <->  A  e.  B
) )
43pm5.32ri 642 . . . . 5  |-  ( ( x  e.  B  /\  ( x  =  A  /\  ph ) )  <->  ( A  e.  B  /\  (
x  =  A  /\  ph ) ) )
54bicomi 205 . . . 4  |-  ( ( A  e.  B  /\  ( x  =  A  /\  ph ) )  <->  ( x  e.  B  /\  (
x  =  A  /\  ph ) ) )
65baib 911 . . 3  |-  ( x  e.  B  ->  (
( A  e.  B  /\  ( x  =  A  /\  ph ) )  <-> 
( x  =  A  /\  ph ) ) )
76rexbiia 2865 . 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 3147 . . 3  |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ps )
)
109pm5.32i 641 . 2  |-  ( ( A  e.  B  /\  E. x  e.  B  ( x  =  A  /\  ph ) )  <->  ( A  e.  B  /\  ps )
)
111, 7, 103bitr3i 278 1  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ( A  e.  B  /\  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   E.wrex 2715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-rex 2720  df-v 3024
This theorem is referenced by:  marypha2lem2  7903  txkgen  20609  ceqsrexv2  30308  eq0rabdioph  35531
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