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Theorem reusv1 4379
Description: Two ways to express single-valuedness of a class expression  C ( y ). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
reusv1  |-  ( E. y  e.  B  ph  ->  ( E! x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  <->  E. x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) ) )
Distinct variable groups:    x, A    x, B    x, C    ph, x    x, y
Allowed substitution hints:    ph( y)    A( y)    B( y)    C( y)

Proof of Theorem reusv1
StepHypRef Expression
1 nfra1 2466 . . . 4  |-  F/ y A. y  e.  B  ( ph  ->  x  =  C )
21nfmo 2019 . . 3  |-  F/ y E* x A. y  e.  B  ( ph  ->  x  =  C )
3 rsp 2480 . . . . . . . 8  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  ( y  e.  B  ->  ( ph  ->  x  =  C ) ) )
43impd 252 . . . . . . 7  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  ( (
y  e.  B  /\  ph )  ->  x  =  C ) )
54com12 30 . . . . . 6  |-  ( ( y  e.  B  /\  ph )  ->  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  x  =  C ) )
65alrimiv 1846 . . . . 5  |-  ( ( y  e.  B  /\  ph )  ->  A. x
( A. y  e.  B  ( ph  ->  x  =  C )  ->  x  =  C )
)
7 moeq 2859 . . . . 5  |-  E* x  x  =  C
8 moim 2063 . . . . 5  |-  ( A. x ( A. y  e.  B  ( ph  ->  x  =  C )  ->  x  =  C )  ->  ( E* x  x  =  C  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) ) )
96, 7, 8mpisyl 1422 . . . 4  |-  ( ( y  e.  B  /\  ph )  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) )
109ex 114 . . 3  |-  ( y  e.  B  ->  ( ph  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) ) )
112, 10rexlimi 2542 . 2  |-  ( E. y  e.  B  ph  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) )
12 mormo 2642 . 2  |-  ( E* x A. y  e.  B  ( ph  ->  x  =  C )  ->  E* x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) )
13 reu5 2643 . . 3  |-  ( E! x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  <->  ( E. x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  /\  E* x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) ) )
1413rbaib 906 . 2  |-  ( E* x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  ->  ( E! x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  <->  E. x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) ) )
1511, 12, 143syl 17 1  |-  ( E. y  e.  B  ph  ->  ( E! x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  <->  E. x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   E*wmo 2000   A.wral 2416   E.wrex 2417   E!wreu 2418   E*wrmo 2419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-v 2688
This theorem is referenced by: (None)
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