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Theorem eqrdav 2455
 Description: Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
Hypotheses
Ref Expression
eqrdav.1
eqrdav.2
eqrdav.3
Assertion
Ref Expression
eqrdav
Distinct variable groups:   ,   ,   ,

Proof of Theorem eqrdav
StepHypRef Expression
1 eqrdav.1 . . . 4
2 eqrdav.3 . . . . . 6
32biimpd 207 . . . . 5
43impancom 440 . . . 4
51, 4mpd 15 . . 3
6 eqrdav.2 . . . 4
72biimprd 223 . . . . 5
87impancom 440 . . . 4
96, 8mpd 15 . . 3
105, 9impbida 832 . 2
1110eqrdv 2454 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818 This theorem is referenced by:  boxcutc  7532  supminf  11198  f1omvdconj  16471  fmucndlem  20794  ballotlemsima  28454 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-cleq 2449
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