Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdinex1 Structured version   GIF version

Theorem bdinex1 6466
Description: Bounded version of inex1 3864. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdinex1.bd BOUNDED B
bdinex1.1 A V
Assertion
Ref Expression
bdinex1 (AB) V

Proof of Theorem bdinex1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdinex1.1 . . . 4 A V
2 bdinex1.bd . . . . . 6 BOUNDED B
32bdeli 6421 . . . . 5 BOUNDED y B
43bdzfauscl 6460 . . . 4 (A V → xy(y x ↔ (y A y B)))
51, 4ax-mp 7 . . 3 xy(y x ↔ (y A y B))
6 dfcleq 2017 . . . . 5 (x = (AB) ↔ y(y xy (AB)))
7 elin 3102 . . . . . . 7 (y (AB) ↔ (y A y B))
87bibi2i 216 . . . . . 6 ((y xy (AB)) ↔ (y x ↔ (y A y B)))
98albii 1339 . . . . 5 (y(y xy (AB)) ↔ y(y x ↔ (y A y B)))
106, 9bitri 173 . . . 4 (x = (AB) ↔ y(y x ↔ (y A y B)))
1110exbii 1480 . . 3 (x x = (AB) ↔ xy(y x ↔ (y A y B)))
125, 11mpbir 134 . 2 x x = (AB)
1312issetri 2541 1 (AB) V
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1226   = wceq 1228  wex 1363   wcel 1375  Vcvv 2534  cin 2892  BOUNDED wbdc 6415
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 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-bdsep 6455
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-in 2900  df-bdc 6416
This theorem is referenced by:  bdinex2  6467  bdinex1g  6468  bdpeano5  6510
  Copyright terms: Public domain W3C validator