Theorem disjeq1 3973

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 3202	. . 3 |- ( A = B -> B C_ A )
2		disjss1 3972	. . 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 3201	. . 3 |- ( A = B -> A C_ B )
5		disjss1 3972	. . 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 128	1 |- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   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-rmo 2456  df-in 3127  df-ss 3134  df-disj 3967

This theorem is referenced by:  disjeq1d  3974