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Theorem mob 3023
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 2872 . . . . 5  |-  ( B  e.  D  ->  B  e.  _V )
2 nfcv 2494 . . . . . . . 8  |-  F/_ x A
3 nfv 1619 . . . . . . . . . 10  |-  F/ x  B  e.  _V
4 nfmo1 2220 . . . . . . . . . 10  |-  F/ x E* x ph
5 nfv 1619 . . . . . . . . . 10  |-  F/ x ps
63, 4, 5nf3an 1832 . . . . . . . . 9  |-  F/ x
( B  e.  _V  /\ 
E* x ph  /\  ps )
7 nfv 1619 . . . . . . . . 9  |-  F/ x
( A  =  B  <->  ch )
86, 7nfim 1815 . . . . . . . 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 2364 . . . . . . . . . 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 3021 . . . . . . . 8  |-  ( ( B  e.  _V  /\  E* x ph  /\  ph )  ->  ( x  =  B  <->  ch ) )
162, 8, 13, 15vtoclgf 2918 . . . . . . 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 1642    e. wcel 1710   E*wmo 2210   _Vcvv 2864
This theorem is referenced by:  moi  3024  rmob  3155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866
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