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Theorem iinconstm 3858
Description: Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Jim Kingdon, 19-Dec-2018.)
Assertion
Ref Expression
iinconstm (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem iinconstm
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vex 2715 . . . 4 𝑧 ∈ V
2 eliin 3854 . . . 4 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
31, 2ax-mp 5 . . 3 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
4 r19.3rmv 3484 . . 3 (∃𝑦 𝑦𝐴 → (𝑧𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
53, 4bitr4id 198 . 2 (∃𝑦 𝑦𝐴 → (𝑧 𝑥𝐴 𝐵𝑧𝐵))
65eqrdv 2155 1 (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1335  wex 1472  wcel 2128  wral 2435  Vcvv 2712   ciin 3850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-iin 3852
This theorem is referenced by:  iin0imm  4129
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