Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  negcmpprcal2 Unicode version

Theorem negcmpprcal2 25049
Description: Negation of a complex predicated inequality. (Contributed by FL, 31-Jul-2009.)
Assertion
Ref Expression
negcmpprcal2  |-  ( -. 
E. x  e.  A  A. y  e.  B  C  =/=  D  <->  A. x  e.  A  E. y  e.  B  C  =  D )

Proof of Theorem negcmpprcal2
StepHypRef Expression
1 rexnal 2567 . . . 4  |-  ( E. y  e.  B  -.  C  =/=  D  <->  -.  A. y  e.  B  C  =/=  D )
21bicomi 193 . . 3  |-  ( -. 
A. y  e.  B  C  =/=  D  <->  E. y  e.  B  -.  C  =/=  D )
32ralbii 2580 . 2  |-  ( A. x  e.  A  -.  A. y  e.  B  C  =/=  D  <->  A. x  e.  A  E. y  e.  B  -.  C  =/=  D
)
4 ralnex 2566 . 2  |-  ( A. x  e.  A  -.  A. y  e.  B  C  =/=  D  <->  -.  E. x  e.  A  A. y  e.  B  C  =/=  D )
5 nne 2463 . . . 4  |-  ( -.  C  =/=  D  <->  C  =  D )
65rexbii 2581 . . 3  |-  ( E. y  e.  B  -.  C  =/=  D  <->  E. y  e.  B  C  =  D )
76ralbii 2580 . 2  |-  ( A. x  e.  A  E. y  e.  B  -.  C  =/=  D  <->  A. x  e.  A  E. y  e.  B  C  =  D )
83, 4, 73bitr3i 266 1  |-  ( -. 
E. x  e.  A  A. y  e.  B  C  =/=  D  <->  A. x  e.  A  E. y  e.  B  C  =  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1632    =/= wne 2459   A.wral 2556   E.wrex 2557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-ne 2461  df-ral 2561  df-rex 2562
  Copyright terms: Public domain W3C validator