Theorem disjeq2dv 4368

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 2825	. 2 |- ( ph -> A. x e. A B = C )
3		disjeq2 4367	. 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 1370   e. wcel 1758   A.wral 2795  Disj wdisj 4363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-ral 2800  df-rmo 2803  df-in 3436  df-ss 3443  df-disj 4364

This theorem is referenced by:  disjeq12d  4372  iunmbl  21160  uniioovol  21185  voliunnfl  28576