MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjeq12d Structured version   Visualization version   GIF version

Theorem disjeq12d 4821
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disjeq1d.1 (𝜑𝐴 = 𝐵)
disjeq12d.1 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
disjeq12d (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem disjeq12d
StepHypRef Expression
1 disjeq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21disjeq1d 4820 . 2 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
3 disjeq12d.1 . . . 4 (𝜑𝐶 = 𝐷)
43adantr 473 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
54disjeq2dv 4817 . 2 (𝜑 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 𝐷))
62, 5bitrd 271 1 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  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-ral 3095  df-rmo 3098  df-in 3777  df-ss 3784  df-disj 4813
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator