Users' Mathboxes Mathbox for Gino Giotto < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjeq12dv Structured version   Visualization version   GIF version

Theorem disjeq12dv 36194
Description: Equality theorem for disjoint collection. Deduction version. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
disjeq12dv.1 (𝜑𝐴 = 𝐵)
disjeq12dv.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
disjeq12dv (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem disjeq12dv
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 disjeq12dv.1 . . . . . . . 8 (𝜑𝐴 = 𝐵)
21eleq2d 2826 . . . . . . 7 (𝜑 → (𝑥𝐴𝑥𝐵))
32anbi1d 631 . . . . . 6 (𝜑 → ((𝑥𝐴𝑡𝐶) ↔ (𝑥𝐵𝑡𝐶)))
43mobidv 2548 . . . . 5 (𝜑 → (∃*𝑥(𝑥𝐴𝑡𝐶) ↔ ∃*𝑥(𝑥𝐵𝑡𝐶)))
5 df-rmo 3379 . . . . 5 (∃*𝑥𝐴 𝑡𝐶 ↔ ∃*𝑥(𝑥𝐴𝑡𝐶))
6 df-rmo 3379 . . . . 5 (∃*𝑥𝐵 𝑡𝐶 ↔ ∃*𝑥(𝑥𝐵𝑡𝐶))
74, 5, 63bitr4g 314 . . . 4 (𝜑 → (∃*𝑥𝐴 𝑡𝐶 ↔ ∃*𝑥𝐵 𝑡𝐶))
87albidv 1920 . . 3 (𝜑 → (∀𝑡∃*𝑥𝐴 𝑡𝐶 ↔ ∀𝑡∃*𝑥𝐵 𝑡𝐶))
9 df-disj 5109 . . 3 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑡∃*𝑥𝐴 𝑡𝐶)
10 df-disj 5109 . . 3 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑡∃*𝑥𝐵 𝑡𝐶)
118, 9, 103bitr4g 314 . 2 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
12 disjeq12dv.2 . . . 4 (𝜑𝐶 = 𝐷)
1312adantr 480 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
1413disjeq2dv 5113 . 2 (𝜑 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 𝐷))
1511, 14bitrd 279 1 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  ∃*wmo 2537  ∃*wrmo 3378  Disj wdisj 5108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2539  df-cleq 2728  df-clel 2815  df-ral 3061  df-rmo 3379  df-ss 3967  df-disj 5109
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator