Theorem disjeq2 3970

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 3202	. . . 4 |- ( B = C -> C C_ B )
2	1	ralimi 2533	. . 3 |- ( A. x e. A B = C -> A. x e. A C C_ B )
3		disjss2 3969	. . 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 3201	. . . 4 |- ( B = C -> B C_ C )
6	5	ralimi 2533	. . 3 |- ( A. x e. A B = C -> A. x e. A B C_ C )
7		disjss2 3969	. . 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 1348   A.wral 2448   C_ wss 3121  Disj wdisj 3966

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-rmo 2456  df-in 3127  df-ss 3134  df-disj 3967

This theorem is referenced by:  disjeq2dv  3971