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Theorem cover2g 29806
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 4253 . . . 4  |-  ( b  =  B  ->  U. b  =  U. B )
2 cover2g.1 . . . 4  |-  A  = 
U. B
31, 2syl6eqr 2526 . . 3  |-  ( b  =  B  ->  U. b  =  A )
4 rexeq 3059 . . 3  |-  ( b  =  B  ->  ( E. y  e.  b 
( x  e.  y  /\  ph )  <->  E. y  e.  B  ( x  e.  y  /\  ph )
) )
53, 4raleqbidv 3072 . 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 4013 . . 3  |-  ( b  =  B  ->  ~P b  =  ~P B
)
73eqeq2d 2481 . . . 4  |-  ( b  =  B  ->  ( U. z  =  U. b 
<-> 
U. z  =  A ) )
87anbi1d 704 . . 3  |-  ( b  =  B  ->  (
( U. z  = 
U. b  /\  A. y  e.  z  ph ) 
<->  ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
96, 8rexeqbidv 3073 . 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 3116 . . 3  |-  b  e. 
_V
11 eqid 2467 . . 3  |-  U. b  =  U. b
1210, 11cover2 29805 . 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 3172 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   ~Pcpw 4010   U.cuni 4245
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-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-in 3483  df-ss 3490  df-pw 4012  df-uni 4246
This theorem is referenced by: (None)
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