Theorem disjeq1 4066

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

Assertion

Ref	Expression
disjeq1	|- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem disjeq1

Step	Hyp	Ref	Expression
1		eqimss2 3279	. . 3 |- ( A = B -> B C_ A )
2		disjss1 4065	. . 3 |- ( B C_ A -> (Disj x e. A C -> Disj x e. B C ) )
3	1, 2	syl 14	. 2 |- ( A = B -> (Disj x e. A C -> Disj x e. B C ) )
4		eqimss 3278	. . 3 |- ( A = B -> A C_ B )
5		disjss1 4065	. . 3 |- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )
6	4, 5	syl 14	. 2 |- ( A = B -> (Disj x e. B C -> Disj x e. A C ) )
7	3, 6	impbid 129	1 |- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   C_ wss 3197  Disj wdisj 4059

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-rmo 2516  df-in 3203  df-ss 3210  df-disj 4060

This theorem is referenced by:  disjeq1d  4067