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Theorem eqreu 3205
Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypothesis
Ref Expression
eqreu.1  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
eqreu  |-  ( ( B  e.  A  /\  ps  /\  A. x  e.  A  ( ph  ->  x  =  B ) )  ->  E! x  e.  A  ph )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem eqreu
StepHypRef Expression
1 ralbiim 2899 . . . . 5  |-  ( A. x  e.  A  ( ph 
<->  x  =  B )  <-> 
( A. x  e.  A  ( ph  ->  x  =  B )  /\  A. x  e.  A  ( x  =  B  ->  ph ) ) )
2 eqreu.1 . . . . . . 7  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
32ceqsralv 3052 . . . . . 6  |-  ( B  e.  A  ->  ( A. x  e.  A  ( x  =  B  ->  ph )  <->  ps )
)
43anbi2d 708 . . . . 5  |-  ( B  e.  A  ->  (
( A. x  e.  A  ( ph  ->  x  =  B )  /\  A. x  e.  A  ( x  =  B  ->  ph ) )  <->  ( A. x  e.  A  ( ph  ->  x  =  B )  /\  ps )
) )
51, 4syl5bb 260 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ph  <->  x  =  B
)  <->  ( A. x  e.  A  ( ph  ->  x  =  B )  /\  ps ) ) )
6 reu6i 3204 . . . . 5  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
76ex 435 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ph  <->  x  =  B
)  ->  E! x  e.  A  ph ) )
85, 7sylbird 238 . . 3  |-  ( B  e.  A  ->  (
( A. x  e.  A  ( ph  ->  x  =  B )  /\  ps )  ->  E! x  e.  A  ph ) )
983impib 1203 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph  ->  x  =  B )  /\  ps )  ->  E! x  e.  A  ph )
1093com23 1211 1  |-  ( ( B  e.  A  /\  ps  /\  A. x  e.  A  ( ph  ->  x  =  B ) )  ->  E! x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2714   E!wreu 2716
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-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-reu 2721  df-v 3024
This theorem is referenced by:  uzwo3  11210  frmdup3  16594  frgpup3  17371  neiptopreu  20091  ufileu  20876  mirreu  24651  lmireu  24774  symgfcoeu  28560
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