Theorem disjeq12d 4629

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 4628	. 2 |- ( ph -> (Disj x e. A C <-> Disj x e. B C ) )
3		disjeq12d.1	. . . 4 |- ( ph -> C = D )
4	3	adantr 481	. . 3 |- ( ( ph /\ x e. B ) -> C = D )
5	4	disjeq2dv 4625	. 2 |- ( ph -> (Disj x e. B C <-> Disj x e. B D ) )
6	2, 5	bitrd 268	1 |- ( ph -> (Disj x e. A C <-> Disj x e. B D ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990  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-ral 2917  df-rmo 2920  df-in 3581  df-ss 3588  df-disj 4621

This theorem is referenced by: (None)