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Theorem eqrdav 2139
 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 143 . . . . 5
43impancom 258 . . . 4
51, 4mpd 13 . . 3
6 eqrdav.2 . . . 4
72exbiri 380 . . . . . 6
87com23 78 . . . . 5
98imp 123 . . . 4
106, 9mpd 13 . . 3
115, 10impbida 586 . 2
1211eqrdv 2138 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1332   wcel 1481 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-17 1507  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-cleq 2133 This theorem is referenced by:  supminfex  9439  fzdifsuc  9912
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