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Theorem ineq1 3450
 Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
Assertion
Ref Expression
ineq1 (A = B → (AC) = (BC))

Proof of Theorem ineq1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2414 . . . 4 (A = B → (x Ax B))
21anbi1d 685 . . 3 (A = B → ((x A x C) ↔ (x B x C)))
3 elin 3219 . . 3 (x (AC) ↔ (x A x C))
4 elin 3219 . . 3 (x (BC) ↔ (x B x C))
52, 3, 43bitr4g 279 . 2 (A = B → (x (AC) ↔ x (BC)))
65eqrdv 2351 1 (A = B → (AC) = (BC))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3208 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213 This theorem is referenced by:  ineq2  3451  ineq12  3452  ineq1i  3453  ineq1d  3456  unineq  3505  dfrab3ss  3533  intprg  3960  eladdci  4399  addcass  4415  nndisjeq  4429  reseq1  4928  brdisjg  5821  qsdisj  5995
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