Theorem disjeq1 4076

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 3283	. . 3 |- ( A = B -> B C_ A )
2		disjss1 4075	. . 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 3282	. . 3 |- ( A = B -> A C_ B )
5		disjss1 4075	. . 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 1398   C_ wss 3201  Disj wdisj 4069

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-rmo 2519  df-in 3207  df-ss 3214  df-disj 4070

This theorem is referenced by:  disjeq1d  4077