New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  eqrdav Unicode version

Theorem eqrdav 2352
 Description: Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)
Hypotheses
Ref Expression
eqrdav.1
eqrdav.2
eqrdav.3
Assertion
Ref Expression
eqrdav
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem eqrdav
StepHypRef Expression
1 eqrdav.1 . . . 4
2 eqrdav.3 . . . . . 6
32biimpd 198 . . . . 5
43impancom 427 . . . 4
51, 4mpd 14 . . 3
6 eqrdav.2 . . . 4
72exbiri 605 . . . . . 6
87com23 72 . . . . 5
98imp 418 . . . 4
106, 9mpd 14 . . 3
115, 10impbida 805 . 2
1211eqrdv 2351 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710 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-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2346 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator