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Theorem eqreu 3248
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 2950 . . . . 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 3097 . . . . . 6  |-  ( B  e.  A  ->  ( A. x  e.  A  ( x  =  B  ->  ph )  <->  ps )
)
43anbi2d 703 . . . . 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 3247 . . . . 5  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph 
<->  x  =  B ) )  ->  E! x  e.  A  ph )
76ex 434 . . . 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 1186 . 2  |-  ( ( B  e.  A  /\  A. x  e.  A  (
ph  ->  x  =  B )  /\  ps )  ->  E! x  e.  A  ph )
1093com23 1194 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   E!wreu 2797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-reu 2802  df-v 3070
This theorem is referenced by:  uzwo3  11049  frmdup3  15646  frgpup3  16379  neiptopreu  18853  ufileu  19608  mirreu  23194
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