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Theorem eqreu 2970
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 2693 . . . . 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 2828 . . . . . 6  |-  ( B  e.  A  ->  ( A. x  e.  A  ( x  =  B  ->  ph )  <->  ps )
)
43anbi2d 684 . . . . 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 248 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ph  <->  x  =  B
)  <->  ( A. x  e.  A  ( ph  ->  x  =  B )  /\  ps ) ) )
6 reu6i 2969 . . . . 5  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
76ex 423 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ph  <->  x  =  B
)  ->  E! x  e.  A  ph ) )
85, 7sylbird 226 . . 3  |-  ( B  e.  A  ->  (
( A. x  e.  A  ( ph  ->  x  =  B )  /\  ps )  ->  E! x  e.  A  ph ) )
983impib 1149 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph  ->  x  =  B )  /\  ps )  ->  E! x  e.  A  ph )
1093com23 1157 1  |-  ( ( B  e.  A  /\  ps  /\  A. x  e.  A  ( ph  ->  x  =  B ) )  ->  E! x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E!wreu 2558
This theorem is referenced by:  uzwo3  10327  frmdup3  14504  frgpup3  15103  ufileu  17630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-v 2803
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