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Theorem reu6i 3276
Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
reu6i  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem reu6i
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2458 . . . . 5  |-  ( y  =  B  ->  (
x  =  y  <->  x  =  B ) )
21bibi2d 318 . . . 4  |-  ( y  =  B  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  B ) ) )
32ralbidv 2882 . . 3  |-  ( y  =  B  ->  ( A. x  e.  A  ( ph  <->  x  =  y
)  <->  A. x  e.  A  ( ph  <->  x  =  B
) ) )
43rspcev 3196 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E. y  e.  A  A. x  e.  A  ( ph  <->  x  =  y ) )
5 reu6 3274 . 2  |-  ( E! x  e.  A  ph  <->  E. y  e.  A  A. x  e.  A  ( ph 
<->  x  =  y ) )
64, 5sylibr 212 1  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   E!wreu 2795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-reu 2800  df-v 3097
This theorem is referenced by:  eqreu  3277  riota5f  6267  negeu  9815  creur  10536  creui  10537  reuccats1  12685  lublecl  15493  dfod2  16460  lmieu  24022
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