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Theorem iinconst 4662
 Description: Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Mario Carneiro, 6-Feb-2015.)
Assertion
Ref Expression
iinconst (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iinconst
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.3rzv 4203 . . 3 (𝐴 ≠ ∅ → (𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
2 vex 3352 . . . 4 𝑦 ∈ V
3 eliin 4657 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
42, 3ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
51, 4syl6rbbr 279 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
65eqrdv 2768 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1630   ∈ wcel 2144   ≠ wne 2942  ∀wral 3060  Vcvv 3349  ∅c0 4061  ∩ ciin 4653 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-v 3351  df-dif 3724  df-nul 4062  df-iin 4655 This theorem is referenced by:  iin0  4967  ptbasfi  21604
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