Theorem disjeq2 4039

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 3256	. . . 4 |- ( B = C -> C C_ B )
2	1	ralimi 2571	. . 3 |- ( A. x e. A B = C -> A. x e. A C C_ B )
3		disjss2 4038	. . 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 3255	. . . 4 |- ( B = C -> B C_ C )
6	5	ralimi 2571	. . 3 |- ( A. x e. A B = C -> A. x e. A B C_ C )
7		disjss2 4038	. . 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 1373   A.wral 2486   C_ wss 3174  Disj wdisj 4035

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-rmo 2494  df-in 3180  df-ss 3187  df-disj 4036

This theorem is referenced by:  disjeq2dv  4040