Theorem disjeq12d 3923

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 3922	. 2 |- ( ph -> (Disj x e. A C <-> Disj x e. B C ) )
3		disjeq12d.1	. . . 4 |- ( ph -> C = D )
4	3	adantr 274	. . 3 |- ( ( ph /\ x e. B ) -> C = D )
5	4	disjeq2dv 3919	. 2 |- ( ph -> (Disj x e. B C <-> Disj x e. B D ) )
6	2, 5	bitrd 187	1 |- ( ph -> (Disj x e. A C <-> Disj x e. B D ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481  Disj wdisj 3914

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-rmo 2425  df-in 3082  df-ss 3089  df-disj 3915

This theorem is referenced by: (None)