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Theorem iinconstm 3657
 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 y A x A 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 3306 . . 3 (y y A → (z Bx A z B))
2 vex 2554 . . . 4 z V
3 eliin 3653 . . . 4 (z V → (z x A Bx A z B))
42, 3ax-mp 7 . . 3 (z x A Bx A z B)
51, 4syl6rbbr 188 . 2 (y y A → (z x A Bz B))
65eqrdv 2035 1 (y y A x A B = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  Vcvv 2551  ∩ ciin 3649 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-iin 3651 This theorem is referenced by:  iin0imm  3912
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