Theorem disjeq12d 4044

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 4043	. 2 |- ( ph -> (Disj x e. A C <-> Disj x e. B C ) )
3		disjeq12d.1	. . . 4 |- ( ph -> C = D )
4	3	adantr 276	. . 3 |- ( ( ph /\ x e. B ) -> C = D )
5	4	disjeq2dv 4040	. 2 |- ( ph -> (Disj x e. B C <-> Disj x e. B D ) )
6	2, 5	bitrd 188	1 |- ( ph -> (Disj x e. A C <-> Disj x e. B D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178  Disj wdisj 4035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-rmo 2494  df-in 3180  df-ss 3187  df-disj 4036

This theorem is referenced by: (None)