ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mob2 Unicode version

Theorem mob2 2837
Description: Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)
Hypothesis
Ref Expression
moi2.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
mob2  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem mob2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp3 968 . . 3  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ph )
2 moi2.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2syl5ibcom 154 . 2  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  ->  ps )
)
4 nfs1v 1892 . . . . . . . 8  |-  F/ x [ y  /  x ] ph
5 sbequ12 1729 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
64, 5mo4f 2037 . . . . . . 7  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
7 sp 1473 . . . . . . 7  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
86, 7sylbi 120 . . . . . 6  |-  ( E* x ph  ->  A. y
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
9 nfv 1493 . . . . . . . . . 10  |-  F/ x ps
109, 2sbhypf 2709 . . . . . . . . 9  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  ps ) )
1110anbi2d 459 . . . . . . . 8  |-  ( y  =  A  ->  (
( ph  /\  [ y  /  x ] ph ) 
<->  ( ph  /\  ps ) ) )
12 eqeq2 2127 . . . . . . . 8  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
1311, 12imbi12d 233 . . . . . . 7  |-  ( y  =  A  ->  (
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  ps )  ->  x  =  A )
) )
1413spcgv 2747 . . . . . 6  |-  ( A  e.  B  ->  ( A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ( ph  /\  ps )  ->  x  =  A ) ) )
158, 14syl5 32 . . . . 5  |-  ( A  e.  B  ->  ( E* x ph  ->  (
( ph  /\  ps )  ->  x  =  A ) ) )
1615imp 123 . . . 4  |-  ( ( A  e.  B  /\  E* x ph )  -> 
( ( ph  /\  ps )  ->  x  =  A ) )
1716expd 256 . . 3  |-  ( ( A  e.  B  /\  E* x ph )  -> 
( ph  ->  ( ps 
->  x  =  A
) ) )
18173impia 1163 . 2  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( ps  ->  x  =  A ) )
193, 18impbid 128 1  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947   A.wal 1314    = wceq 1316    e. wcel 1465   [wsb 1720   E*wmo 1978
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  moi2  2838  mob  2839
  Copyright terms: Public domain W3C validator