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Theorem reu6i 3287
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 2469 . . . . 5  |-  ( y  =  B  ->  (
x  =  y  <->  x  =  B ) )
21bibi2d 316 . . . 4  |-  ( y  =  B  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  B ) ) )
32ralbidv 2893 . . 3  |-  ( y  =  B  ->  ( A. x  e.  A  ( ph  <->  x  =  y
)  <->  A. x  e.  A  ( ph  <->  x  =  B
) ) )
43rspcev 3207 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E. y  e.  A  A. x  e.  A  ( ph  <->  x  =  y ) )
5 reu6 3285 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   E!wreu 2806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-reu 2811  df-v 3108
This theorem is referenced by:  eqreu  3288  riota5f  6256  negeu  9801  creur  10525  creui  10526  reuccats1  12697  lublecl  15818  dfod2  16785  lmieu  24351  esum2dlem  28321  reuccatpfxs1  32662
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