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Theorem symdifeq1 3248
 Description: Equality law for intersection. (Contributed by SF, 11-Jan-2015.)
Assertion
Ref Expression
symdifeq1 (A = B → (AC) = (BC))

Proof of Theorem symdifeq1
StepHypRef Expression
1 difeq1 3246 . . . 4 (A = B → (A C) = (B C))
21compleqd 3245 . . 3 (A = B → ∼ (A C) = ∼ (B C))
3 difeq2 3247 . . . 4 (A = B → (C A) = (C B))
43compleqd 3245 . . 3 (A = B → ∼ (C A) = ∼ (C B))
52, 4nineq12d 3242 . 2 (A = B → ( ∼ (A C) ⩃ ∼ (C A)) = ( ∼ (B C) ⩃ ∼ (C B)))
6 df-symdif 3216 . . 3 (AC) = ((A C) ∪ (C A))
7 df-un 3214 . . 3 ((A C) ∪ (C A)) = ( ∼ (A C) ⩃ ∼ (C A))
86, 7eqtri 2373 . 2 (AC) = ( ∼ (A C) ⩃ ∼ (C A))
9 df-symdif 3216 . . 3 (BC) = ((B C) ∪ (C B))
10 df-un 3214 . . 3 ((B C) ∪ (C B)) = ( ∼ (B C) ⩃ ∼ (C B))
119, 10eqtri 2373 . 2 (BC) = ( ∼ (B C) ⩃ ∼ (C B))
125, 8, 113eqtr4g 2410 1 (A = B → (AC) = (BC))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ⩃ cnin 3204   ∼ ccompl 3205   ∖ cdif 3206   ∪ cun 3207   ⊕ csymdif 3209 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  df-un 3214  df-dif 3215  df-symdif 3216 This theorem is referenced by:  symdifeq12  3250  symdifeq1i  3251  symdifeq1d  3254
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