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Theorem iinconstm 3691
 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 3337 . . 3 (∃𝑦 𝑦𝐴 → (𝑧𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
2 vex 2575 . . . 4 𝑧 ∈ V
3 eliin 3687 . . . 4 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
42, 3ax-mp 7 . . 3 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
51, 4syl6rbbr 192 . 2 (∃𝑦 𝑦𝐴 → (𝑧 𝑥𝐴 𝐵𝑧𝐵))
65eqrdv 2052 1 (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   = wceq 1257  ∃wex 1395   ∈ wcel 1407  ∀wral 2321  Vcvv 2572  ∩ ciin 3683 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036 This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-v 2574  df-iin 3685 This theorem is referenced by:  iin0imm  3946
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