Theorem disjeq12d 3967

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 3966	. 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 3963	. 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 1343   e. wcel 2136  Disj wdisj 3958

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2448  df-rmo 2451  df-in 3121  df-ss 3128  df-disj 3959

This theorem is referenced by: (None)