Theorem disjeq1d 4628

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

Hypothesis

Ref	Expression
disjeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
disjeq1d	|- ( ph -> (Disj x e. A C <-> Disj x e. B C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   C( x)

Proof of Theorem disjeq1d

Step	Hyp	Ref	Expression
1		disjeq1d.1	. 2 |- ( ph -> A = B )
2		disjeq1 4627	. 2 |- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )
3	1, 2	syl 17	1 |- ( ph -> (Disj x e. A C <-> Disj x e. B C ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-rmo 2920  df-in 3581  df-ss 3588  df-disj 4621

This theorem is referenced by:  disjeq12d  4629  disjxiun  4649  disjxiunOLD  4650  disjdifprg  29388  disjdifprg2  29389  disjun0  29408  measxun2  30273  measssd  30278  meadjun  40679