Theorem cover2g 33509

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.

Step	Hyp	Ref	Expression
1		unieq 4444	. . . 4 |- ( b = B -> U. b = U. B )
2		cover2g.1	. . . 4 |- A = U. B
3	1, 2	syl6eqr 2674	. . 3 |- ( b = B -> U. b = A )
4		rexeq 3139	. . 3 |- ( b = B -> ( E. y e. b ( x e. y /\ ph ) <-> E. y e. B ( x e. y /\ ph ) ) )
5	3, 4	raleqbidv 3152	. 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 4161	. . 3 |- ( b = B -> ~P b = ~P B )
7	3	eqeq2d 2632	. . . 4 |- ( b = B -> ( U. z = U. b <-> U. z = A ) )
8	7	anbi1d 741	. . 3 |- ( b = B -> ( ( U. z = U. b /\ A. y e. z ph ) <-> ( U. z = A /\ A. y e. z ph ) ) )
9	6, 8	rexeqbidv 3153	. 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 3203	. . 3 |- b e. _V
11		eqid 2622	. . 3 |- U. b = U. b
12	10, 11	cover2 33508	. 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 ) )
13	5, 9, 12	vtoclbg 3267	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 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ~Pcpw 4158   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by: (None)