Theorem disjeq12d 3754

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

Hypotheses

Ref	Expression
disjeq1d.1	|- ( ph -> A = B )
disjeq12d.1	|- ( ph -> C = D )

Assertion

Ref	Expression
disjeq12d	|- ( ph -> (Disj x e. A C <-> Disj x e. B D ) )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hints:   C( x)   D( x)

Proof of Theorem disjeq12d

Step	Hyp	Ref	Expression
1		disjeq1d.1	. . 3 |- ( ph -> A = B )
2	1	disjeq1d 3753	. 2 |- ( ph -> (Disj x e. A C <-> Disj x e. B C ) )
3		disjeq12d.1	. . . 4 |- ( ph -> C = D )
4	3	adantr 261	. . 3 |- ( ( ph /\ x e. B ) -> C = D )
5	4	disjeq2dv 3750	. 2 |- ( ph -> (Disj x e. B C <-> Disj x e. B D ) )
6	2, 5	bitrd 177	1 |- ( ph -> (Disj x e. A C <-> Disj x e. B D ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393  Disj wdisj 3745

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-rmo 2314  df-in 2924  df-ss 2931  df-disj 3746

This theorem is referenced by: (None)