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Theorem llyeq 17196
Description: Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llyeq  |-  ( A  =  B  -> Locally  A  = Locally  B )

Proof of Theorem llyeq
Dummy variables  j  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2344 . . . . . 6  |-  ( A  =  B  ->  (
( jt  u )  e.  A  <->  ( jt  u )  e.  B
) )
21anbi2d 684 . . . . 5  |-  ( A  =  B  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  <->  ( y  e.  u  /\  ( jt  u )  e.  B ) ) )
32rexbidv 2564 . . . 4  |-  ( A  =  B  ->  ( E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A )  <->  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  B ) ) )
432ralbidv 2585 . . 3  |-  ( A  =  B  ->  ( A. x  e.  j  A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A )  <->  A. x  e.  j  A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  B ) ) )
54rabbidv 2780 . 2  |-  ( A  =  B  ->  { j  e.  Top  |  A. x  e.  j  A. y  e.  x  E. u  e.  ( j  i^i  ~P x ) ( y  e.  u  /\  ( jt  u )  e.  A
) }  =  {
j  e.  Top  |  A. x  e.  j  A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  B ) } )
6 df-lly 17192 . 2  |- Locally  A  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A ) }
7 df-lly 17192 . 2  |- Locally  B  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  B ) }
85, 6, 73eqtr4g 2340 1  |-  ( A  =  B  -> Locally  A  = Locally  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151   ~Pcpw 3625  (class class class)co 5858   ↾t crest 13325   Topctop 16631  Locally clly 17190
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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-rex 2549  df-rab 2552  df-lly 17192
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