Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdinex1g Structured version   GIF version

Theorem bdinex1g 7271
 Description: Bounded version of inex1g 3867. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdinex1g.bd BOUNDED B
Assertion
Ref Expression
bdinex1g (A 𝑉 → (AB) V)

Proof of Theorem bdinex1g
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ineq1 3108 . . 3 (x = A → (xB) = (AB))
21eleq1d 2088 . 2 (x = A → ((xB) V ↔ (AB) V))
3 bdinex1g.bd . . 3 BOUNDED B
4 vex 2538 . . 3 x V
53, 4bdinex1 7269 . 2 (xB) V
62, 5vtoclg 2590 1 (A 𝑉 → (AB) V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1228   ∈ wcel 1374  Vcvv 2535   ∩ cin 2893  BOUNDED wbdc 7214 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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bdsep 7258 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-bdc 7215 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator