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Theorem iinconstm 3910
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 2755 . . . 4 𝑧 ∈ V
2 eliin 3906 . . . 4 (𝑧 ∈ V → (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
31, 2ax-mp 5 . . 3 (𝑧 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑧𝐵)
4 r19.3rmv 3528 . . 3 (∃𝑦 𝑦𝐴 → (𝑧𝐵 ↔ ∀𝑥𝐴 𝑧𝐵))
53, 4bitr4id 199 . 2 (∃𝑦 𝑦𝐴 → (𝑧 𝑥𝐴 𝐵𝑧𝐵))
65eqrdv 2187 1 (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wex 1503  wcel 2160  wral 2468  Vcvv 2752   ciin 3902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-iin 3904
This theorem is referenced by:  iin0imm  4183
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