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Theorem iinconstm 3640
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 3290 . . 3 (y y A → (z Bx A z B))
2 vex 2538 . . . 4 z V
3 eliin 3636 . . . 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 2020 1 (y y A x A B = B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228  wex 1362   wcel 1374  wral 2284  Vcvv 2535   ciin 3632
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-iin 3634
This theorem is referenced by:  iin0imm  3895
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