Theorem disjeq2dv 3964

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 2539	. 2 |- ( ph -> A. x e. A B = C )
3		disjeq2 3963	. 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 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444  Disj wdisj 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-rmo 2452  df-in 3122  df-ss 3129  df-disj 3960

This theorem is referenced by:  disjeq12d  3968