Theorem disjeq2dv 4412

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 2857	. 2 |- ( ph -> A. x e. A B = C )
3		disjeq2 4411	. 2 |- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )
4	2, 3	syl 16	1 |- ( ph -> (Disj x e. A B <-> Disj x e. A C ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   A.wral 2793  Disj wdisj 4407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-ral 2798  df-rmo 2801  df-in 3468  df-ss 3475  df-disj 4408

This theorem is referenced by:  disjeq12d  4416  iunmbl  21836  uniioovol  21861  voliunnfl  30033