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Theorem ssint 3621
Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
ssint (A Bx B Ax)
Distinct variable groups:   x,A   x,B

Proof of Theorem ssint
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfss3 2929 . 2 (A By A y B)
2 vex 2554 . . . 4 y V
32elint2 3612 . . 3 (y Bx B y x)
43ralbii 2324 . 2 (y A y By A x B y x)
5 ralcom 2467 . . 3 (y A x B y xx B y A y x)
6 dfss3 2929 . . . 4 (Axy A y x)
76ralbii 2324 . . 3 (x B Axx B y A y x)
85, 7bitr4i 176 . 2 (y A x B y xx B Ax)
91, 4, 83bitri 195 1 (A Bx B Ax)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1390  wral 2300  wss 2911   cint 3605
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-bnd 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-in 2918  df-ss 2925  df-int 3606
This theorem is referenced by:  ssintab  3622  ssintub  3623  iinpw  3732  trint  3859  fintm  5016  bj-ssom  8989
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