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Theorem ineq12 3452
 Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
Assertion
Ref Expression
ineq12 ((A = B C = D) → (AC) = (BD))

Proof of Theorem ineq12
StepHypRef Expression
1 ineq1 3450 . 2 (A = B → (AC) = (BC))
2 ineq2 3451 . 2 (C = D → (BC) = (BD))
31, 2sylan9eq 2405 1 ((A = B C = D) → (AC) = (BD))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  ineq12i  3455  ineq12d  3458  ineqan12d  3459  fnun  5189  fvun1  5379  fntxp  5804  endisj  6051  ncdisjeq  6148  letc  6231
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