Theorem disjeq2 3905

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 3147	. . . 4 |- ( B = C -> C C_ B )
2	1	ralimi 2493	. . 3 |- ( A. x e. A B = C -> A. x e. A C C_ B )
3		disjss2 3904	. . 3 |- ( A. x e. A C C_ B -> (Disj x e. A B -> Disj x e. A C ) )
4	2, 3	syl 14	. 2 |- ( A. x e. A B = C -> (Disj x e. A B -> Disj x e. A C ) )
5		eqimss 3146	. . . 4 |- ( B = C -> B C_ C )
6	5	ralimi 2493	. . 3 |- ( A. x e. A B = C -> A. x e. A B C_ C )
7		disjss2 3904	. . 3 |- ( A. x e. A B C_ C -> (Disj x e. A C -> Disj x e. A B ) )
8	6, 7	syl 14	. 2 |- ( A. x e. A B = C -> (Disj x e. A C -> Disj x e. A B ) )
9	4, 8	impbid 128	1 |- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   A.wral 2414   C_ wss 3066  Disj wdisj 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-rmo 2422  df-in 3072  df-ss 3079  df-disj 3902

This theorem is referenced by:  disjeq2dv  3906