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Theorem mob2 3114
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 959 . . 3  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ph )
2 moi2.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2syl5ibcom 212 . 2  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  ->  ps )
)
4 nfs1v 2182 . . . . . . . 8  |-  F/ x [ y  /  x ] ph
5 sbequ12 1944 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
64, 5mo4f 2313 . . . . . . 7  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
7 sp 1763 . . . . . . 7  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
86, 7sylbi 188 . . . . . 6  |-  ( E* x ph  ->  A. y
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
9 nfv 1629 . . . . . . . . . 10  |-  F/ x ps
109, 2sbhypf 3001 . . . . . . . . 9  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  ps ) )
1110anbi2d 685 . . . . . . . 8  |-  ( y  =  A  ->  (
( ph  /\  [ y  /  x ] ph ) 
<->  ( ph  /\  ps ) ) )
12 eqeq2 2445 . . . . . . . 8  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
1311, 12imbi12d 312 . . . . . . 7  |-  ( y  =  A  ->  (
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  ps )  ->  x  =  A )
) )
1413spcgv 3036 . . . . . 6  |-  ( A  e.  B  ->  ( A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ( ph  /\  ps )  ->  x  =  A ) ) )
158, 14syl5 30 . . . . 5  |-  ( A  e.  B  ->  ( E* x ph  ->  (
( ph  /\  ps )  ->  x  =  A ) ) )
1615imp 419 . . . 4  |-  ( ( A  e.  B  /\  E* x ph )  -> 
( ( ph  /\  ps )  ->  x  =  A ) )
1716exp3a 426 . . 3  |-  ( ( A  e.  B  /\  E* x ph )  -> 
( ph  ->  ( ps 
->  x  =  A
) ) )
18173impia 1150 . 2  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( ps  ->  x  =  A ) )
193, 18impbid 184 1  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652   [wsb 1658    e. wcel 1725   E*wmo 2282
This theorem is referenced by:  moi2  3115  mob  3116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958
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