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Theorem eqreu 3288
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 2986 . . . . 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 3135 . . . . . 6  |-  ( B  e.  A  ->  ( A. x  e.  A  ( x  =  B  ->  ph )  <->  ps )
)
43anbi2d 701 . . . . 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 257 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ph  <->  x  =  B
)  <->  ( A. x  e.  A  ( ph  ->  x  =  B )  /\  ps ) ) )
6 reu6i 3287 . . . . 5  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
76ex 432 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ph  <->  x  =  B
)  ->  E! x  e.  A  ph ) )
85, 7sylbird 235 . . 3  |-  ( B  e.  A  ->  (
( A. x  e.  A  ( ph  ->  x  =  B )  /\  ps )  ->  E! x  e.  A  ph ) )
983impib 1192 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph  ->  x  =  B )  /\  ps )  ->  E! x  e.  A  ph )
1093com23 1200 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 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   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-3an 973  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:  uzwo3  11178  frmdup3  16237  frgpup3  16998  neiptopreu  19804  ufileu  20589  mirreu  24249  lmireu  24360
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