Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  intabs Structured version   Unicode version

Theorem intabs 4363
 Description: Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
Hypotheses
Ref Expression
intabs.1
intabs.2
intabs.3
Assertion
Ref Expression
intabs
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem intabs
StepHypRef Expression
1 sseq1 3371 . . . . . 6
2 intabs.2 . . . . . 6
31, 2anbi12d 693 . . . . 5
4 intabs.3 . . . . 5
53, 4intmin3 4080 . . . 4
6 intnex 4359 . . . . 5
7 ssv 3370 . . . . . 6
8 sseq2 3372 . . . . . 6
97, 8mpbiri 226 . . . . 5
106, 9sylbi 189 . . . 4
115, 10pm2.61i 159 . . 3
12 intabs.1 . . . . 5
1312cbvabv 2557 . . . 4
1413inteqi 4056 . . 3
1511, 14sseqtr4i 3383 . 2
16 simpr 449 . . . 4
1716ss2abi 3417 . . 3
18 intss 4073 . . 3
1917, 18ax-mp 8 . 2
2015, 19eqssi 3366 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  cab 2424  cvv 2958   wss 3322  cint 4052 This theorem is referenced by:  dfnn3  10016 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-int 4053
 Copyright terms: Public domain W3C validator