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Theorem negcmpprcal2 24946
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 2554 . . . 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 2567 . 2  |-  ( A. x  e.  A  -.  A. y  e.  B  C  =/=  D  <->  A. x  e.  A  E. y  e.  B  -.  C  =/=  D
)
4 ralnex 2553 . 2  |-  ( A. x  e.  A  -.  A. y  e.  B  C  =/=  D  <->  -.  E. x  e.  A  A. y  e.  B  C  =/=  D )
5 nne 2450 . . . 4  |-  ( -.  C  =/=  D  <->  C  =  D )
65rexbii 2568 . . 3  |-  ( E. y  e.  B  -.  C  =/=  D  <->  E. y  e.  B  C  =  D )
76ralbii 2567 . 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 1623    =/= wne 2446   A.wral 2543   E.wrex 2544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-ne 2448  df-ral 2548  df-rex 2549
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