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Theorem cover2g 26416
Description: Two ways of expressing the statement "there is a cover of  A by elements of  B such that for each set in the cover,  ph." Note that  ph and  x must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)
Hypothesis
Ref Expression
cover2g.1  |-  A  = 
U. B
Assertion
Ref Expression
cover2g  |-  ( B  e.  C  ->  ( A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )  <->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
Distinct variable groups:    ph, x, z   
x, B, y, z   
x, A, z
Allowed substitution hints:    ph( y)    A( y)    C( x, y, z)

Proof of Theorem cover2g
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 unieq 4024 . . . 4  |-  ( b  =  B  ->  U. b  =  U. B )
2 cover2g.1 . . . 4  |-  A  = 
U. B
31, 2syl6eqr 2486 . . 3  |-  ( b  =  B  ->  U. b  =  A )
4 rexeq 2905 . . 3  |-  ( b  =  B  ->  ( E. y  e.  b 
( x  e.  y  /\  ph )  <->  E. y  e.  B  ( x  e.  y  /\  ph )
) )
53, 4raleqbidv 2916 . 2  |-  ( b  =  B  ->  ( A. x  e.  U. b E. y  e.  b 
( x  e.  y  /\  ph )  <->  A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )
) )
6 pweq 3802 . . 3  |-  ( b  =  B  ->  ~P b  =  ~P B
)
73eqeq2d 2447 . . . 4  |-  ( b  =  B  ->  ( U. z  =  U. b 
<-> 
U. z  =  A ) )
87anbi1d 686 . . 3  |-  ( b  =  B  ->  (
( U. z  = 
U. b  /\  A. y  e.  z  ph ) 
<->  ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
96, 8rexeqbidv 2917 . 2  |-  ( b  =  B  ->  ( E. z  e.  ~P  b ( U. z  =  U. b  /\  A. y  e.  z  ph ) 
<->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
10 vex 2959 . . 3  |-  b  e. 
_V
11 eqid 2436 . . 3  |-  U. b  =  U. b
1210, 11cover2 26415 . 2  |-  ( A. x  e.  U. b E. y  e.  b 
( x  e.  y  /\  ph )  <->  E. z  e.  ~P  b ( U. z  =  U. b  /\  A. y  e.  z 
ph ) )
135, 9, 12vtoclbg 3012 1  |-  ( B  e.  C  ->  ( A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )  <->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   ~Pcpw 3799   U.cuni 4015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-in 3327  df-ss 3334  df-pw 3801  df-uni 4016
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