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Theorem mob 3232
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 3066 . . . . 5  |-  ( B  e.  D  ->  B  e.  _V )
2 nfv 1772 . . . . . . . . . 10  |-  F/ x  B  e.  _V
3 nfmo1 2321 . . . . . . . . . 10  |-  F/ x E* x ph
4 nfv 1772 . . . . . . . . . 10  |-  F/ x ps
52, 3, 4nf3an 2024 . . . . . . . . 9  |-  F/ x
( B  e.  _V  /\ 
E* x ph  /\  ps )
6 nfv 1772 . . . . . . . . 9  |-  F/ x
( A  =  B  <->  ch )
75, 6nfim 2014 . . . . . . . 8  |-  F/ x
( ( B  e. 
_V  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
8 moi.1 . . . . . . . . . 10  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
983anbi3d 1354 . . . . . . . . 9  |-  ( x  =  A  ->  (
( B  e.  _V  /\ 
E* x ph  /\  ph )  <->  ( B  e. 
_V  /\  E* x ph  /\  ps ) ) )
10 eqeq1 2466 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
1110bibi1d 325 . . . . . . . . 9  |-  ( x  =  A  ->  (
( x  =  B  <->  ch )  <->  ( A  =  B  <->  ch ) ) )
129, 11imbi12d 326 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( B  e. 
_V  /\  E* x ph  /\  ph )  -> 
( x  =  B  <->  ch ) )  <->  ( ( B  e.  _V  /\  E* x ph  /\  ps )  ->  ( A  =  B  <->  ch ) ) ) )
13 moi.2 . . . . . . . . 9  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
1413mob2 3230 . . . . . . . 8  |-  ( ( B  e.  _V  /\  E* x ph  /\  ph )  ->  ( x  =  B  <->  ch ) )
157, 12, 14vtoclg1f 3118 . . . . . . 7  |-  ( A  e.  C  ->  (
( B  e.  _V  /\ 
E* x ph  /\  ps )  ->  ( A  =  B  <->  ch )
) )
1615com12 32 . . . . . 6  |-  ( ( B  e.  _V  /\  E* x ph  /\  ps )  ->  ( A  e.  C  ->  ( A  =  B  <->  ch ) ) )
17163expib 1218 . . . . 5  |-  ( B  e.  _V  ->  (
( E* x ph  /\ 
ps )  ->  ( A  e.  C  ->  ( A  =  B  <->  ch )
) ) )
181, 17syl 17 . . . 4  |-  ( B  e.  D  ->  (
( E* x ph  /\ 
ps )  ->  ( A  e.  C  ->  ( A  =  B  <->  ch )
) ) )
1918com3r 82 . . 3  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ( E* x ph  /\ 
ps )  ->  ( A  =  B  <->  ch )
) ) )
2019imp 435 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) ) )
21203impib 1213 1  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   E*wmo 2311   _Vcvv 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059
This theorem is referenced by:  moi  3233  rmob  3371
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