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Theorem mob2 3160
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 990 . . 3  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ph )
2 moi2.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2syl5ibcom 220 . 2  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  ->  ps )
)
4 nfs1v 2142 . . . . . . . 8  |-  F/ x [ y  /  x ] ph
5 sbequ12 1936 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
64, 5mo4f 2316 . . . . . . 7  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
7 sp 1794 . . . . . . 7  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
86, 7sylbi 195 . . . . . 6  |-  ( E* x ph  ->  A. y
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
9 nfv 1673 . . . . . . . . . 10  |-  F/ x ps
109, 2sbhypf 3040 . . . . . . . . 9  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  ps ) )
1110anbi2d 703 . . . . . . . 8  |-  ( y  =  A  ->  (
( ph  /\  [ y  /  x ] ph ) 
<->  ( ph  /\  ps ) ) )
12 eqeq2 2452 . . . . . . . 8  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
1311, 12imbi12d 320 . . . . . . 7  |-  ( y  =  A  ->  (
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  ps )  ->  x  =  A )
) )
1413spcgv 3078 . . . . . 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 429 . . . 4  |-  ( ( A  e.  B  /\  E* x ph )  -> 
( ( ph  /\  ps )  ->  x  =  A ) )
1716expd 436 . . 3  |-  ( ( A  e.  B  /\  E* x ph )  -> 
( ph  ->  ( ps 
->  x  =  A
) ) )
18173impia 1184 . 2  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( ps  ->  x  =  A ) )
193, 18impbid 191 1  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369   [wsb 1700    e. wcel 1756   E*wmo 2254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2995
This theorem is referenced by:  moi2  3161  mob  3162
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