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

Theorem cbviindavw 36206
Description: Change bound variable in indexed intersections. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbviindavw.1 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
Assertion
Ref Expression
cbviindavw (𝜑 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶)
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbviindavw
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cbviindavw.1 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
21eleq2d 2823 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑡𝐵𝑡𝐶))
32cbvraldva 3235 . . 3 (𝜑 → (∀𝑥𝐴 𝑡𝐵 ↔ ∀𝑦𝐴 𝑡𝐶))
43abbidv 2804 . 2 (𝜑 → {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵} = {𝑡 ∣ ∀𝑦𝐴 𝑡𝐶})
5 df-iin 5001 . 2 𝑥𝐴 𝐵 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵}
6 df-iin 5001 . 2 𝑦𝐴 𝐶 = {𝑡 ∣ ∀𝑦𝐴 𝑡𝐶}
74, 5, 63eqtr4g 2798 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1535  wcel 2104  {cab 2710  wral 3057   ciin 4999
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-ral 3058  df-iin 5001
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator