Theorem disjeq2 4624

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjeq2	|- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )

Proof of Theorem disjeq2

Step	Hyp	Ref	Expression
1		eqimss2 3658	. . . 4 |- ( B = C -> C C_ B )
2	1	ralimi 2952	. . 3 |- ( A. x e. A B = C -> A. x e. A C C_ B )
3		disjss2 4623	. . 3 |- ( A. x e. A C C_ B -> (Disj x e. A B -> Disj x e. A C ) )
4	2, 3	syl 17	. 2 |- ( A. x e. A B = C -> (Disj x e. A B -> Disj x e. A C ) )
5		eqimss 3657	. . . 4 |- ( B = C -> B C_ C )
6	5	ralimi 2952	. . 3 |- ( A. x e. A B = C -> A. x e. A B C_ C )
7		disjss2 4623	. . 3 |- ( A. x e. A B C_ C -> (Disj x e. A C -> Disj x e. A B ) )
8	6, 7	syl 17	. 2 |- ( A. x e. A B = C -> (Disj x e. A C -> Disj x e. A B ) )
9	4, 8	impbid 202	1 |- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   A.wral 2912   C_ wss 3574  Disj wdisj 4620

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-rmo 2920  df-in 3581  df-ss 3588  df-disj 4621

This theorem is referenced by:  disjeq2dv  4625  voliun  23322  carsgclctunlem2  30381  mblfinlem2  33447  voliunnfl  33453