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Theorem iinin2m 3699
Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iinin2m (x x A x A (B𝐶) = (B x A 𝐶))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝐶(x)

Proof of Theorem iinin2m
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 r19.28mv 3293 . . . 4 (x x A → (x A (y B y 𝐶) ↔ (y B x A y 𝐶)))
2 elin 3103 . . . . 5 (y (B𝐶) ↔ (y B y 𝐶))
32ralbii 2308 . . . 4 (x A y (B𝐶) ↔ x A (y B y 𝐶))
4 vex 2538 . . . . . 6 y V
5 eliin 3636 . . . . . 6 (y V → (y x A 𝐶x A y 𝐶))
64, 5ax-mp 7 . . . . 5 (y x A 𝐶x A y 𝐶)
76anbi2i 433 . . . 4 ((y B y x A 𝐶) ↔ (y B x A y 𝐶))
81, 3, 73bitr4g 212 . . 3 (x x A → (x A y (B𝐶) ↔ (y B y x A 𝐶)))
9 eliin 3636 . . . 4 (y V → (y x A (B𝐶) ↔ x A y (B𝐶)))
104, 9ax-mp 7 . . 3 (y x A (B𝐶) ↔ x A y (B𝐶))
11 elin 3103 . . 3 (y (B x A 𝐶) ↔ (y B y x A 𝐶))
128, 10, 113bitr4g 212 . 2 (x x A → (y x A (B𝐶) ↔ y (B x A 𝐶)))
1312eqrdv 2020 1 (x x A x A (B𝐶) = (B x A 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  wral 2284  Vcvv 2535  cin 2893   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-in 2901  df-iin 3634
This theorem is referenced by:  iinin1m  3700
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