Theorem disjeq2dv 4391

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 2813	. 2 |- ( ph -> A. x e. A B = C )
3		disjeq2 4390	. 2 |- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )
4	2, 3	syl 17	1 |- ( ph -> (Disj x e. A B <-> Disj x e. A C ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1454   e. wcel 1897   A.wral 2748  Disj wdisj 4386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-ral 2753  df-rmo 2756  df-in 3422  df-ss 3429  df-disj 4387

This theorem is referenced by:  disjeq12d  4395  iunmbl  22554  uniioovol  22584  carsggect  29198  voliunnfl  32028  nnfoctbdjlem  38330  meadjiun  38341