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Theorem iinconstm 3734
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 r19.3rmv 3368 . . 3 (∃𝑦 𝑦𝐴 → (𝑧𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
2 vex 2622 . . . 4 𝑧 ∈ V
3 eliin 3730 . . . 4 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
42, 3ax-mp 7 . . 3 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
51, 4syl6rbbr 197 . 2 (∃𝑦 𝑦𝐴 → (𝑧 𝑥𝐴 𝐵𝑧𝐵))
65eqrdv 2086 1 (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wex 1426  wcel 1438  wral 2359  Vcvv 2619   ciin 3726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-iin 3728
This theorem is referenced by:  iin0imm  3995
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