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Theorem iineq1 4471
 Description: Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
iineq1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iineq1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 raleq 3115 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 𝑦𝐶 ↔ ∀𝑥𝐵 𝑦𝐶))
21abbidv 2728 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐶})
3 df-iin 4458 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
4 df-iin 4458 . 2 𝑥𝐵 𝐶 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐶}
52, 3, 43eqtr4g 2669 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-iin 4458 This theorem is referenced by:  iinrab2  4519  iinvdif  4528  riin0  4530  iin0  4765  xpriindi  5180  cmpfi  21021  ptbasfi  21194  fclsval  21622  taylfval  23917  polvalN  34209  iineq1d  38295
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