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Theorem iineq2 5012
Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iineq2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Proof of Theorem iineq2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2830 . . . . 5 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 3083 . . . 4 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 ralbi 3103 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
42, 3syl 17 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
54abbidv 2808 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵} = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶})
6 df-iin 4994 . 2 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
7 df-iin 4994 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
85, 6, 73eqtr4g 2802 1 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {cab 2714  wral 3061   ciin 4992
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 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-iin 4994
This theorem is referenced by:  iineq2i  5014  iineq2d  5015  iineq2dv  5017  firest  17477  iincld  23047  elrfirn2  42707
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