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Theorem iineq1 4909
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 3390 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 𝑦𝐶 ↔ ∀𝑥𝐵 𝑦𝐶))
21abbidv 2885 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐶})
3 df-iin 4895 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐶}
4 df-iin 4895 . 2 𝑥𝐵 𝐶 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐶}
52, 3, 43eqtr4g 2881 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {cab 2799  wral 3126   ciin 4893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-ral 3131  df-iin 4895
This theorem is referenced by:  iinrab2  4965  iinvdif  4975  riin0  4977  iin0  5234  xpriindi  5680  cmpfi  21992  ptbasfi  22165  fclsval  22592  taylfval  24933  polvalN  37087  iineq1d  41543
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