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Theorem eqreu 2929
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 2611 . . . . 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 2768 . . . . . 6  |-  ( B  e.  A  ->  ( A. x  e.  A  ( x  =  B  ->  ph )  <->  ps )
)
43anbi2d 464 . . . . 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, 4bitrid 192 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ph  <->  x  =  B
)  <->  ( A. x  e.  A  ( ph  ->  x  =  B )  /\  ps ) ) )
6 reu6i 2928 . . . . 5  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
76ex 115 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ph  <->  x  =  B
)  ->  E! x  e.  A  ph ) )
85, 7sylbird 170 . . 3  |-  ( B  e.  A  ->  (
( A. x  e.  A  ( ph  ->  x  =  B )  /\  ps )  ->  E! x  e.  A  ph ) )
983impib 1201 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph  ->  x  =  B )  /\  ps )  ->  E! x  e.  A  ph )
1093com23 1209 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    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   E!wreu 2457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2739
This theorem is referenced by: (None)
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