ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reusv1 Unicode version

Theorem reusv1 4218
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 2372 . . . 4  |-  F/ y A. y  e.  B  ( ph  ->  x  =  C )
21nfmo 1936 . . 3  |-  F/ y E* x A. y  e.  B  ( ph  ->  x  =  C )
3 rsp 2386 . . . . . . . 8  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  ( y  e.  B  ->  ( ph  ->  x  =  C ) ) )
43impd 246 . . . . . . 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 1770 . . . . 5  |-  ( ( y  e.  B  /\  ph )  ->  A. x
( A. y  e.  B  ( ph  ->  x  =  C )  ->  x  =  C )
)
7 moeq 2739 . . . . 5  |-  E* x  x  =  C
8 moim 1980 . . . . 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 1351 . . . 4  |-  ( ( y  e.  B  /\  ph )  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) )
109ex 112 . . 3  |-  ( y  e.  B  ->  ( ph  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) ) )
112, 10rexlimi 2443 . 2  |-  ( E. y  e.  B  ph  ->  E* x A. y  e.  B  ( ph  ->  x  =  C ) )
12 mormo 2538 . 2  |-  ( E* x A. y  e.  B  ( ph  ->  x  =  C )  ->  E* x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) )
13 reu5 2539 . . 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 841 . 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 101    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   E*wmo 1917   A.wral 2323   E.wrex 2324   E!wreu 2325   E*wrmo 2326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-v 2576
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator