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Theorem nineq1 3234
 Description: Equality law for anti-intersection. (Contributed by SF, 11-Jan-2015.)
Assertion
Ref Expression
nineq1 (A = B → (AC) = (BC))

Proof of Theorem nineq1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2414 . . . 4 (A = B → (x Ax B))
21nanbi1d 1301 . . 3 (A = B → ((x A x C) ↔ (x B x C)))
32abbidv 2467 . 2 (A = B → {x (x A x C)} = {x (x B x C)})
4 df-nin 3211 . 2 (AC) = {x (x A x C)}
5 df-nin 3211 . 2 (BC) = {x (x B x C)}
63, 4, 53eqtr4g 2410 1 (A = B → (AC) = (BC))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊼ wnan 1287   = wceq 1642   ∈ wcel 1710  {cab 2339   ⩃ cnin 3204 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-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-nin 3211 This theorem is referenced by:  nineq2  3235  nineq12  3236  nineq1i  3237  nineq1d  3240  difeq1  3246  ninexg  4097
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