Theorem iundif1 33383

Description: Indexed union of class difference with the subtrahend held constant. (Contributed by Brendan Leahy, 6-Aug-2018.)

Assertion

Ref	Expression
iundif1	|- U_ x e. A ( B \ C ) = ( U_ x e. A B \ C )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem iundif1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.41v 3089	. . . 4 |- ( E. x e. A ( y e. B /\ -. y e. C ) <-> ( E. x e. A y e. B /\ -. y e. C ) )
2		eldif 3584	. . . . 5 |- ( y e. ( B \ C ) <-> ( y e. B /\ -. y e. C ) )
3	2	rexbii 3041	. . . 4 |- ( E. x e. A y e. ( B \ C ) <-> E. x e. A ( y e. B /\ -. y e. C ) )
4		eliun 4524	. . . . 5 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
5	4	anbi1i 731	. . . 4 |- ( ( y e. U_ x e. A B /\ -. y e. C ) <-> ( E. x e. A y e. B /\ -. y e. C ) )
6	1, 3, 5	3bitr4i 292	. . 3 |- ( E. x e. A y e. ( B \ C ) <-> ( y e. U_ x e. A B /\ -. y e. C ) )
7		eliun 4524	. . 3 |- ( y e. U_ x e. A ( B \ C ) <-> E. x e. A y e. ( B \ C ) )
8		eldif 3584	. . 3 |- ( y e. ( U_ x e. A B \ C ) <-> ( y e. U_ x e. A B /\ -. y e. C ) )
9	6, 7, 8	3bitr4i 292	. 2 |- ( y e. U_ x e. A ( B \ C ) <-> y e. ( U_ x e. A B \ C ) )
10	9	eqriv 2619	1 |- U_ x e. A ( B \ C ) = ( U_ x e. A B \ C )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   U_ciun 4520

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

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-v 3202  df-dif 3577  df-iun 4522

This theorem is referenced by: (None)