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

Theorem eqrdav 2465
Description: Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
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 )
72biimprd 223 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
x  e.  B  ->  x  e.  A )
)
87impancom 440 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  e.  C  ->  x  e.  A )
)
96, 8mpd 15 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
105, 9impbida 830 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
1110eqrdv 2464 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-cleq 2459
This theorem is referenced by:  boxcutc  7524  supminf  11181  f1omvdconj  16344  fmucndlem  20662  ballotlemsima  28279
  Copyright terms: Public domain W3C validator