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

Theorem llyeq 17525
 Description: Equality theorem for the Locally predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llyeq Locally Locally

Proof of Theorem llyeq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2496 . . . . . 6 t t
21anbi2d 685 . . . . 5 t t
32rexbidv 2718 . . . 4 t t
432ralbidv 2739 . . 3 t t
54rabbidv 2940 . 2 t t
6 df-lly 17521 . 2 Locally t
7 df-lly 17521 . 2 Locally t
85, 6, 73eqtr4g 2492 1 Locally Locally
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  crab 2701   cin 3311  cpw 3791  (class class class)co 6073   ↾t crest 13640  ctop 16950  Locally clly 17519 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-ral 2702  df-rex 2703  df-rab 2706  df-lly 17521
 Copyright terms: Public domain W3C validator