Theorem disjeq2dv 3771

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

Hypothesis

Ref	Expression
disjeq2dv.1	|- ( ( ph /\ x e. A ) -> B = C )

Assertion

Ref	Expression
disjeq2dv	|- ( ph -> (Disj x e. A B <-> Disj x e. A C ) )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   B( x)   C( x)

Proof of Theorem disjeq2dv

Step	Hyp	Ref	Expression
1		disjeq2dv.1	. . 3 |- ( ( ph /\ x e. A ) -> B = C )
2	1	ralrimiva 2434	. 2 |- ( ph -> A. x e. A B = C )
3		disjeq2 3770	. 2 |- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )
4	2, 3	syl 14	1 |- ( ph -> (Disj x e. A B <-> Disj x e. A C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348  Disj wdisj 3766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-rmo 2356  df-in 2979  df-ss 2986  df-disj 3767

This theorem is referenced by:  disjeq12d  3775