Theorem llyeq 21273

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.

Step	Hyp	Ref	Expression
1		eleq2 2690	. . . . . 6 |- ( A = B -> ( ( j ↾t u ) e. A <-> ( j ↾t u ) e. B ) )
2	1	anbi2d 740	. . . . 5 |- ( A = B -> ( ( y e. u /\ ( j ↾t u ) e. A ) <-> ( y e. u /\ ( j ↾t u ) e. B ) ) )
3	2	rexbidv 3052	. . . 4 |- ( A = B -> ( E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. A ) <-> E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. B ) ) )
4	3	2ralbidv 2989	. . 3 |- ( A = B -> ( A. x e. j A. y e. x E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. A ) <-> A. x e. j A. y e. x E. u e. ( j i^i ~P x ) ( y e. u /\ ( j ↾t u ) e. B ) ) )
5	4	rabbidv 3189	. 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 /\ ( j ↾t 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 /\ ( j ↾t u ) e. B ) } )
6		df-lly 21269	. 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 /\ ( j ↾t u ) e. A ) }
7		df-lly 21269	. 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 /\ ( j ↾t u ) e. B ) }
8	5, 6, 7	3eqtr4g 2681	1 |- ( A = B -> Locally A = Locally B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   ~Pcpw 4158  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  Locally clly 21267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-rex 2918  df-rab 2921  df-lly 21269

This theorem is referenced by:  ismntoplly  30069