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Theorem llyeq 20416
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 2502 . . . . . 6  |-  ( A  =  B  ->  (
( jt  u )  e.  A  <->  ( jt  u )  e.  B
) )
21anbi2d 708 . . . . 5  |-  ( A  =  B  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  <->  ( y  e.  u  /\  ( jt  u )  e.  B ) ) )
32rexbidv 2946 . . . 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 2876 . . 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 3079 . 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 20412 . 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 20412 . 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 2495 1  |-  ( A  =  B  -> Locally  A  = Locally  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   {crab 2786    i^i cin 3441   ~Pcpw 3985  (class class class)co 6305   ↾t crest 15278   Topctop 19848  Locally clly 20410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-ral 2787  df-rex 2788  df-rab 2791  df-lly 20412
This theorem is referenced by:  ismntoplly  28668
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