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Theorem iineq2 4694
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 2833 . . . . 5 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 3099 . . . 4 (∀𝑥𝐴 𝐵 = 𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 ralbi 3215 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
42, 3syl 17 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐶))
54abbidv 2884 . 2 (∀𝑥𝐴 𝐵 = 𝐶 → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵} = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶})
6 df-iin 4679 . 2 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
7 df-iin 4679 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
85, 6, 73eqtr4g 2824 1 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wcel 2155  {cab 2751  wral 3055   ciin 4677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-ral 3060  df-iin 4679
This theorem is referenced by:  iineq2i  4696  iineq2d  4697  firest  16359  iincld  21123  elrfirn2  37937
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