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

Theorem injust 3234
Description: Soundness justification theorem for df-in 3235. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
injust  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { y  |  ( y  e.  A  /\  y  e.  B ) }
Distinct variable groups:    x, A    x, B    y, A    y, B

Proof of Theorem injust
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eleq1 2418 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 eleq1 2418 . . . 4  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
31, 2anbi12d 691 . . 3  |-  ( x  =  z  ->  (
( x  e.  A  /\  x  e.  B
)  <->  ( z  e.  A  /\  z  e.  B ) ) )
43cbvabv 2477 . 2  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { z  |  ( z  e.  A  /\  z  e.  B ) }
5 eleq1 2418 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
6 eleq1 2418 . . . 4  |-  ( z  =  y  ->  (
z  e.  B  <->  y  e.  B ) )
75, 6anbi12d 691 . . 3  |-  ( z  =  y  ->  (
( z  e.  A  /\  z  e.  B
)  <->  ( y  e.  A  /\  y  e.  B ) ) )
87cbvabv 2477 . 2  |-  { z  |  ( z  e.  A  /\  z  e.  B ) }  =  { y  |  ( y  e.  A  /\  y  e.  B ) }
94, 8eqtri 2378 1  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { y  |  ( y  e.  A  /\  y  e.  B ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354
  Copyright terms: Public domain W3C validator