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Theorem cover2g 31487
 Description: Two ways of expressing the statement "there is a cover of by elements of such that for each set in the cover, ." Note that and must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)
Hypothesis
Ref Expression
cover2g.1
Assertion
Ref Expression
cover2g
Distinct variable groups:   ,,   ,,,   ,,
Allowed substitution hints:   ()   ()   (,,)

Proof of Theorem cover2g
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 unieq 4199 . . . 4
2 cover2g.1 . . . 4
31, 2syl6eqr 2461 . . 3
4 rexeq 3005 . . 3
53, 4raleqbidv 3018 . 2
6 pweq 3958 . . 3
73eqeq2d 2416 . . . 4
87anbi1d 703 . . 3
96, 8rexeqbidv 3019 . 2
10 vex 3062 . . 3
11 eqid 2402 . . 3
1210, 11cover2 31486 . 2
135, 9, 12vtoclbg 3118 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1405   wcel 1842  wral 2754  wrex 2755  cpw 3955  cuni 4191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-in 3421  df-ss 3428  df-pw 3957  df-uni 4192 This theorem is referenced by: (None)
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