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Theorem ceqsrexbv 2861
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 2627 . 2  |-  ( E. x  e.  B  ( A  e.  B  /\  ( x  =  A  /\  ph ) )  <->  ( A  e.  B  /\  E. x  e.  B  ( x  =  A  /\  ph )
) )
2 eleq1 2233 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32adantr 274 . . . . . 6  |-  ( ( x  =  A  /\  ph )  ->  ( x  e.  B  <->  A  e.  B
) )
43pm5.32ri 452 . . . . 5  |-  ( ( x  e.  B  /\  ( x  =  A  /\  ph ) )  <->  ( A  e.  B  /\  (
x  =  A  /\  ph ) ) )
54bicomi 131 . . . 4  |-  ( ( A  e.  B  /\  ( x  =  A  /\  ph ) )  <->  ( x  e.  B  /\  (
x  =  A  /\  ph ) ) )
65baib 914 . . 3  |-  ( x  e.  B  ->  (
( A  e.  B  /\  ( x  =  A  /\  ph ) )  <-> 
( x  =  A  /\  ph ) ) )
76rexbiia 2485 . 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 2860 . . 3  |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ps )
)
109pm5.32i 451 . 2  |-  ( ( A  e.  B  /\  E. x  e.  B  ( x  =  A  /\  ph ) )  <->  ( A  e.  B  /\  ps )
)
111, 7, 103bitr3i 209 1  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ( A  e.  B  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   E.wrex 2449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732
This theorem is referenced by:  frecsuclem  6385
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