Users' Mathboxes Mathbox for Gino Giotto < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cbviunvw2 Structured version   Visualization version   GIF version

Theorem cbviunvw2 36175
Description: Change bound variable and domain in indexed unions, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbviunvw2.1 (𝑥 = 𝑦𝐶 = 𝐷)
cbviunvw2.2 (𝑥 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
cbviunvw2 𝑥𝐴 𝐶 = 𝑦𝐵 𝐷
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbviunvw2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cbviunvw2.2 . . . 4 (𝑥 = 𝑦𝐴 = 𝐵)
2 cbviunvw2.1 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
32eleq2d 2823 . . . 4 (𝑥 = 𝑦 → (𝑡𝐶𝑡𝐷))
41, 3cbvrexvw2 36170 . . 3 (∃𝑥𝐴 𝑡𝐶 ↔ ∃𝑦𝐵 𝑡𝐷)
54abbii 2805 . 2 {𝑡 ∣ ∃𝑥𝐴 𝑡𝐶} = {𝑡 ∣ ∃𝑦𝐵 𝑡𝐷}
6 df-iun 5000 . 2 𝑥𝐴 𝐶 = {𝑡 ∣ ∃𝑥𝐴 𝑡𝐶}
7 df-iun 5000 . 2 𝑦𝐵 𝐷 = {𝑡 ∣ ∃𝑦𝐵 𝑡𝐷}
85, 6, 73eqtr4i 2771 1 𝑥𝐴 𝐶 = 𝑦𝐵 𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1535  wcel 2104  {cab 2710  wrex 3066   ciun 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rex 3067  df-iun 5000
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator