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Theorem iineq2 4509
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 2687 . . . . 5 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 2947 . . . 4 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 ralbi 3062 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
42, 3syl 17 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
54abbidv 2738 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵} = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶})
6 df-iin 4493 . 2 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
7 df-iin 4493 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
85, 6, 73eqtr4g 2680 1 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  {cab 2607  wral 2907   ciin 4491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-iin 4493
This theorem is referenced by:  iineq2i  4511  iineq2d  4512  firest  16025  iincld  20766  elrfirn2  36774
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