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Theorem eqrdav 2036
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  C
eqrdav.2  C
eqrdav.3  C
Assertion
Ref Expression
eqrdav
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:    C()

Proof of Theorem eqrdav
StepHypRef Expression
1 eqrdav.1 . . . 4  C
2 eqrdav.3 . . . . . 6  C
32biimpd 132 . . . . 5  C
43impancom 247 . . . 4  C
51, 4mpd 13 . . 3
6 eqrdav.2 . . . 4  C
72exbiri 364 . . . . . 6  C
87com23 72 . . . . 5  C
98imp 115 . . . 4  C
106, 9mpd 13 . . 3
115, 10impbida 528 . 2
1211eqrdv 2035 1
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-17 1416  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030
This theorem is referenced by:  fzdifsuc  8693
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