MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nllyeq Unicode version

Theorem nllyeq 17197
Description: Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyeq  |-  ( A  =  B  -> 𝑛Locally  A  = 𝑛Locally  B
)

Proof of Theorem nllyeq
Dummy variables  j  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2344 . . . . 5  |-  ( A  =  B  ->  (
( jt  u )  e.  A  <->  ( jt  u )  e.  B
) )
21rexbidv 2564 . . . 4  |-  ( A  =  B  ->  ( E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A  <->  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  B
) )
322ralbidv 2585 . . 3  |-  ( A  =  B  ->  ( A. x  e.  j  A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A  <->  A. x  e.  j  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  {
y } )  i^i 
~P x ) ( jt  u )  e.  B
) )
43rabbidv 2780 . 2  |-  ( A  =  B  ->  { j  e.  Top  |  A. x  e.  j  A. y  e.  x  E. u  e.  ( (
( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A }  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  B }
)
5 df-nlly 17193 . 2  |- 𝑛Locally  A  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A }
6 df-nlly 17193 . 2  |- 𝑛Locally  B  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  B }
74, 5, 63eqtr4g 2340 1  |-  ( A  =  B  -> 𝑛Locally  A  = 𝑛Locally  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151   ~Pcpw 3625   {csn 3640   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631   neicnei 16834  𝑛Locally cnlly 17191
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-nlly 17193
  Copyright terms: Public domain W3C validator