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Theorem mob 2894
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 2723 . . . . 5  |-  ( B  e.  D  ->  B  e.  _V )
2 nfcv 2299 . . . . . . . 8  |-  F/_ x A
3 nfv 1508 . . . . . . . . . 10  |-  F/ x  B  e.  _V
4 nfmo1 2018 . . . . . . . . . 10  |-  F/ x E* x ph
5 nfv 1508 . . . . . . . . . 10  |-  F/ x ps
63, 4, 5nf3an 1546 . . . . . . . . 9  |-  F/ x
( B  e.  _V  /\ 
E* x ph  /\  ps )
7 nfv 1508 . . . . . . . . 9  |-  F/ x
( A  =  B  <->  ch )
86, 7nfim 1552 . . . . . . . 8  |-  F/ x
( ( B  e. 
_V  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
9 moi.1 . . . . . . . . . 10  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1093anbi3d 1300 . . . . . . . . 9  |-  ( x  =  A  ->  (
( B  e.  _V  /\ 
E* x ph  /\  ph )  <->  ( B  e. 
_V  /\  E* x ph  /\  ps ) ) )
11 eqeq1 2164 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
1211bibi1d 232 . . . . . . . . 9  |-  ( x  =  A  ->  (
( x  =  B  <->  ch )  <->  ( A  =  B  <->  ch ) ) )
1310, 12imbi12d 233 . . . . . . . 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 2892 . . . . . . . 8  |-  ( ( B  e.  _V  /\  E* x ph  /\  ph )  ->  ( x  =  B  <->  ch ) )
162, 8, 13, 15vtoclgf 2770 . . . . . . 7  |-  ( A  e.  C  ->  (
( B  e.  _V  /\ 
E* x ph  /\  ps )  ->  ( A  =  B  <->  ch )
) )
1716com12 30 . . . . . 6  |-  ( ( B  e.  _V  /\  E* x ph  /\  ps )  ->  ( A  e.  C  ->  ( A  =  B  <->  ch ) ) )
18173expib 1188 . . . . 5  |-  ( B  e.  _V  ->  (
( E* x ph  /\ 
ps )  ->  ( A  e.  C  ->  ( A  =  B  <->  ch )
) ) )
191, 18syl 14 . . . 4  |-  ( B  e.  D  ->  (
( E* x ph  /\ 
ps )  ->  ( A  e.  C  ->  ( A  =  B  <->  ch )
) ) )
2019com3r 79 . . 3  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ( E* x ph  /\ 
ps )  ->  ( A  =  B  <->  ch )
) ) )
2120imp 123 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) ) )
22213impib 1183 1  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335   E*wmo 2007    e. wcel 2128   _Vcvv 2712
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714
This theorem is referenced by:  moi  2895  rmob  3029
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