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Theorem reusv1 4616
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 2780 . . . 4  |-  F/ y A. y  e.  B  ( ph  ->  x  =  C )
21nfmo 2326 . . 3  |-  F/ y E* x A. y  e.  B  ( ph  ->  x  =  C )
3 rsp 2765 . . . . . . . 8  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  ( y  e.  B  ->  ( ph  ->  x  =  C ) ) )
43impd 437 . . . . . . 7  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  ( (
y  e.  B  /\  ph )  ->  x  =  C ) )
54com12 32 . . . . . 6  |-  ( ( y  e.  B  /\  ph )  ->  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  x  =  C ) )
65alrimiv 1783 . . . . 5  |-  ( ( y  e.  B  /\  ph )  ->  A. x
( A. y  e.  B  ( ph  ->  x  =  C )  ->  x  =  C )
)
7 moeq 3225 . . . . 5  |-  E* x  x  =  C
8 moim 2358 . . . . 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 21 . . . 4  |-  ( ( y  e.  B  /\  ph )  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) )
109ex 440 . . 3  |-  ( y  e.  B  ->  ( ph  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) ) )
112, 10rexlimi 2880 . 2  |-  ( E. y  e.  B  ph  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) )
12 mormo 3018 . 2  |-  ( E* x A. y  e.  B  ( ph  ->  x  =  C )  ->  E* x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) )
13 reu5 3019 . . 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 922 . 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 18 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452    = wceq 1454    e. wcel 1897   E*wmo 2310   A.wral 2748   E.wrex 2749   E!wreu 2750   E*wrmo 2751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-v 3058
This theorem is referenced by:  cdleme25c  33966  cdleme29c  33987  cdlemefrs29cpre1  34009  cdlemk29-3  34522  cdlemkid5  34546  dihlsscpre  34846  mapdh9a  35402  mapdh9aOLDN  35403
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