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

Theorem disjeq12d 5013
 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 5012 . 2 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
3 disjeq12d.1 . . . 4 (𝜑𝐶 = 𝐷)
43adantr 484 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
54disjeq2dv 5009 . 2 (𝜑 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 𝐷))
62, 5bitrd 282 1 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2115  Disj wdisj 5004 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-mo 2623  df-clab 2800  df-cleq 2814  df-clel 2892  df-ral 3131  df-rmo 3134  df-v 3473  df-in 3917  df-ss 3927  df-disj 5005 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator