Theorem disjeq2 4010

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 3234	. . . 4 |- ( B = C -> C C_ B )
2	1	ralimi 2557	. . 3 |- ( A. x e. A B = C -> A. x e. A C C_ B )
3		disjss2 4009	. . 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 3233	. . . 4 |- ( B = C -> B C_ C )
6	5	ralimi 2557	. . 3 |- ( A. x e. A B = C -> A. x e. A B C_ C )
7		disjss2 4009	. . 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 129	1 |- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   A.wral 2472   C_ wss 3153  Disj wdisj 4006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-rmo 2480  df-in 3159  df-ss 3166  df-disj 4007

This theorem is referenced by:  disjeq2dv  4011