Theorem disjeq1f 29387

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

Hypotheses

Ref	Expression
disjss1f.1	|- F/_ x A
disjss1f.2	|- F/_ x B

Assertion

Ref	Expression
disjeq1f	|- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )

Proof of Theorem disjeq1f

Step	Hyp	Ref	Expression
1		eqimss2 3658	. . 3 |- ( A = B -> B C_ A )
2		disjss1f.2	. . . 4 |- F/_ x B
3		disjss1f.1	. . . 4 |- F/_ x A
4	2, 3	disjss1f 29386	. . 3 |- ( B C_ A -> (Disj x e. A C -> Disj x e. B C ) )
5	1, 4	syl 17	. 2 |- ( A = B -> (Disj x e. A C -> Disj x e. B C ) )
6		eqimss 3657	. . 3 |- ( A = B -> A C_ B )
7	3, 2	disjss1f 29386	. . 3 |- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )
8	6, 7	syl 17	. 2 |- ( A = B -> (Disj x e. B C -> Disj x e. A C ) )
9	5, 8	impbid 202	1 |- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   F/_wnfc 2751   C_ wss 3574  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rmo 2920  df-in 3581  df-ss 3588  df-disj 4621

This theorem is referenced by:  ldgenpisyslem1  30226