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Theorem iineq1 4669
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 3287 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 𝑦𝐶 ↔ ∀𝑥𝐵 𝑦𝐶))
21abbidv 2890 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐶})
3 df-iin 4657 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
4 df-iin 4657 . 2 𝑥𝐵 𝐶 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐶}
52, 3, 43eqtr4g 2830 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {cab 2757  wral 3061   ciin 4655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-iin 4657
This theorem is referenced by:  iinrab2  4717  iinvdif  4726  riin0  4728  iin0  4970  xpriindi  5397  cmpfi  21432  ptbasfi  21605  fclsval  22032  taylfval  24333  polvalN  35713  iineq1d  39788
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