MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjeq1d Structured version   Visualization version   GIF version

Theorem disjeq1d 4820
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypothesis
Ref Expression
disjeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
disjeq1d (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem disjeq1d
StepHypRef Expression
1 disjeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 disjeq1 4819 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
31, 2syl 17 1 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  Disj wdisj 4812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-clab 2787  df-cleq 2793  df-clel 2796  df-rmo 3098  df-in 3777  df-ss 3784  df-disj 4813
This theorem is referenced by:  disjeq12d  4821  disjxiun  4841  disjdifprg  29904  disjdifprg2  29905  disjun0  29924  measxun2  30788  measssd  30793  meadjun  41417
  Copyright terms: Public domain W3C validator