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Theorem mob 2947
Description: Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
Hypotheses
Ref Expression
moi.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
moi.2  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
mob  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
Distinct variable groups:    x, A    x, B    ch, x    ps, x
Allowed substitution hints:    ph( x)    C( x)    D( x)

Proof of Theorem mob
StepHypRef Expression
1 elex 2796 . . . . 5  |-  ( B  e.  D  ->  B  e.  _V )
2 nfcv 2419 . . . . . . . 8  |-  F/_ x A
3 nfv 1605 . . . . . . . . . 10  |-  F/ x  B  e.  _V
4 nfmo1 2154 . . . . . . . . . 10  |-  F/ x E* x ph
5 nfv 1605 . . . . . . . . . 10  |-  F/ x ps
63, 4, 5nf3an 1774 . . . . . . . . 9  |-  F/ x
( B  e.  _V  /\ 
E* x ph  /\  ps )
7 nfv 1605 . . . . . . . . 9  |-  F/ x
( A  =  B  <->  ch )
86, 7nfim 1769 . . . . . . . 8  |-  F/ x
( ( B  e. 
_V  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
9 moi.1 . . . . . . . . . 10  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1093anbi3d 1258 . . . . . . . . 9  |-  ( x  =  A  ->  (
( B  e.  _V  /\ 
E* x ph  /\  ph )  <->  ( B  e. 
_V  /\  E* x ph  /\  ps ) ) )
11 eqeq1 2289 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
1211bibi1d 310 . . . . . . . . 9  |-  ( x  =  A  ->  (
( x  =  B  <->  ch )  <->  ( A  =  B  <->  ch ) ) )
1310, 12imbi12d 311 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( B  e. 
_V  /\  E* x ph  /\  ph )  -> 
( x  =  B  <->  ch ) )  <->  ( ( B  e.  _V  /\  E* x ph  /\  ps )  ->  ( A  =  B  <->  ch ) ) ) )
14 moi.2 . . . . . . . . 9  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
1514mob2 2945 . . . . . . . 8  |-  ( ( B  e.  _V  /\  E* x ph  /\  ph )  ->  ( x  =  B  <->  ch ) )
162, 8, 13, 15vtoclgf 2842 . . . . . . 7  |-  ( A  e.  C  ->  (
( B  e.  _V  /\ 
E* x ph  /\  ps )  ->  ( A  =  B  <->  ch )
) )
1716com12 27 . . . . . 6  |-  ( ( B  e.  _V  /\  E* x ph  /\  ps )  ->  ( A  e.  C  ->  ( A  =  B  <->  ch ) ) )
18173expib 1154 . . . . 5  |-  ( B  e.  _V  ->  (
( E* x ph  /\ 
ps )  ->  ( A  e.  C  ->  ( A  =  B  <->  ch )
) ) )
191, 18syl 15 . . . 4  |-  ( B  e.  D  ->  (
( E* x ph  /\ 
ps )  ->  ( A  e.  C  ->  ( A  =  B  <->  ch )
) ) )
2019com3r 73 . . 3  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ( E* x ph  /\ 
ps )  ->  ( A  =  B  <->  ch )
) ) )
2120imp 418 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) ) )
22213impib 1149 1  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E*wmo 2144   _Vcvv 2788
This theorem is referenced by:  moi  2948  rmob  3079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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