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Theorem eqrdav 2437
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 200 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
x  e.  A  ->  x  e.  B )
)
43impancom 429 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  C  ->  x  e.  B )
)
51, 4mpd 15 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  B )
6 eqrdav.2 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  C )
72exbiri 607 . . . . . 6  |-  ( ph  ->  ( x  e.  C  ->  ( x  e.  B  ->  x  e.  A ) ) )
87com23 75 . . . . 5  |-  ( ph  ->  ( x  e.  B  ->  ( x  e.  C  ->  x  e.  A ) ) )
98imp 420 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  e.  C  ->  x  e.  A )
)
106, 9mpd 15 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
115, 10impbida 807 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
1211eqrdv 2436 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726
This theorem is referenced by:  boxcutc  7107  supminf  10565  fmucndlem  18323  ballotlemsima  24775  f1omvdconj  27368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-cleq 2431
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