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Theorem eqrdav 2114
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  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  C )
eqrdav.2  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  C )
eqrdav.3  |-  ( (
ph  /\  x  e.  C )  ->  (
x  e.  A  <->  x  e.  B ) )
Assertion
Ref Expression
eqrdav  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    C( x)

Proof of Theorem eqrdav
StepHypRef Expression
1 eqrdav.1 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  C )
2 eqrdav.3 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  (
x  e.  A  <->  x  e.  B ) )
32biimpd 143 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
x  e.  A  ->  x  e.  B )
)
43impancom 258 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  C  ->  x  e.  B )
)
51, 4mpd 13 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  B )
6 eqrdav.2 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  C )
72exbiri 377 . . . . . 6  |-  ( ph  ->  ( x  e.  C  ->  ( x  e.  B  ->  x  e.  A ) ) )
87com23 78 . . . . 5  |-  ( ph  ->  ( x  e.  B  ->  ( x  e.  C  ->  x  e.  A ) ) )
98imp 123 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  e.  C  ->  x  e.  A )
)
106, 9mpd 13 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
115, 10impbida 568 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
1211eqrdv 2113 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463
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 1406  ax-gen 1408  ax-17 1489  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108
This theorem is referenced by:  supminfex  9344  fzdifsuc  9812
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