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Theorem iinconstm 3882
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 2733 . . . 4 𝑧 ∈ V
2 eliin 3878 . . . 4 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
31, 2ax-mp 5 . . 3 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
4 r19.3rmv 3505 . . 3 (∃𝑦 𝑦𝐴 → (𝑧𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
53, 4bitr4id 198 . 2 (∃𝑦 𝑦𝐴 → (𝑧 𝑥𝐴 𝐵𝑧𝐵))
65eqrdv 2168 1 (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1348  wex 1485  wcel 2141  wral 2448  Vcvv 2730   ciin 3874
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-iin 3876
This theorem is referenced by:  iin0imm  4154
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