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Theorem iindif2m 3715
Description: Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iindif2m (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 iindif2m
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 r19.28mv 3308 . . . 4 (x x A → (x A (y B ¬ y 𝐶) ↔ (y B x A ¬ y 𝐶)))
2 eldif 2921 . . . . . 6 (y (B𝐶) ↔ (y B ¬ y 𝐶))
32bicomi 123 . . . . 5 ((y B ¬ y 𝐶) ↔ y (B𝐶))
43ralbii 2324 . . . 4 (x A (y B ¬ y 𝐶) ↔ x A y (B𝐶))
5 ralnex 2310 . . . . . 6 (x A ¬ y 𝐶 ↔ ¬ x A y 𝐶)
6 eliun 3652 . . . . . 6 (y x A 𝐶x A y 𝐶)
75, 6xchbinxr 607 . . . . 5 (x A ¬ y 𝐶 ↔ ¬ y x A 𝐶)
87anbi2i 430 . . . 4 ((y B x A ¬ y 𝐶) ↔ (y B ¬ y x A 𝐶))
91, 4, 83bitr3g 211 . . 3 (x x A → (x A y (B𝐶) ↔ (y B ¬ y x A 𝐶)))
10 vex 2554 . . . 4 y V
11 eliin 3653 . . . 4 (y V → (y x A (B𝐶) ↔ x A y (B𝐶)))
1210, 11ax-mp 7 . . 3 (y x A (B𝐶) ↔ x A y (B𝐶))
13 eldif 2921 . . 3 (y (B x A 𝐶) ↔ (y B ¬ y x A 𝐶))
149, 12, 133bitr4g 212 . 2 (x x A → (y x A (B𝐶) ↔ y (B x A 𝐶)))
1514eqrdv 2035 1 (x x A x A (B𝐶) = (B x A 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301  Vcvv 2551  cdif 2908   ciun 3648   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-in1 544  ax-in2 545  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-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-iun 3650  df-iin 3651
This theorem is referenced by: (None)
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