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Theorem llyeq 17212
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 2357 . . . . . 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 2577 . . . 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 2598 . . 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 2793 . 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 17208 . 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 17208 . 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 2353 1  |-  ( A  =  B  -> Locally  A  = Locally  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164   ~Pcpw 3638  (class class class)co 5874   ↾t crest 13341   Topctop 16647  Locally clly 17206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-rex 2562  df-rab 2565  df-lly 17208
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