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Theorem intss 3627
Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
intss (AB B A)

Proof of Theorem intss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imim1 70 . . . . 5 ((y Ay B) → ((y Bx y) → (y Ax y)))
21al2imi 1344 . . . 4 (y(y Ay B) → (y(y Bx y) → y(y Ax y)))
3 vex 2554 . . . . 5 x V
43elint 3612 . . . 4 (x By(y Bx y))
53elint 3612 . . . 4 (x Ay(y Ax y))
62, 4, 53imtr4g 194 . . 3 (y(y Ay B) → (x Bx A))
76alrimiv 1751 . 2 (y(y Ay B) → x(x Bx A))
8 dfss2 2928 . 2 (ABy(y Ay B))
9 dfss2 2928 . 2 ( B Ax(x Bx A))
107, 8, 93imtr4i 190 1 (AB B A)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   wcel 1390  wss 2911   cint 3606
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-v 2553  df-in 2918  df-ss 2925  df-int 3607
This theorem is referenced by: (None)
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