MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqrdav Structured version   Unicode version

Theorem eqrdav 2442
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 207 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
x  e.  A  ->  x  e.  B )
)
43impancom 440 . . . 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 622 . . . . . 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 429 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  e.  C  ->  x  e.  A )
)
106, 9mpd 15 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
115, 10impbida 828 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
1211eqrdv 2441 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-cleq 2436
This theorem is referenced by:  boxcutc  7327  supminf  10963  f1omvdconj  15973  fmucndlem  19888  ballotlemsima  26920
  Copyright terms: Public domain W3C validator