New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  symdifeq2 GIF version

Theorem symdifeq2 3249
 Description: Equality law for intersection. (Contributed by SF, 11-Jan-2015.)
Assertion
Ref Expression
symdifeq2 (A = B → (CA) = (CB))

Proof of Theorem symdifeq2
StepHypRef Expression
1 difeq2 3247 . . . 4 (A = B → (C A) = (C B))
21compleqd 3245 . . 3 (A = B → ∼ (C A) = ∼ (C B))
3 difeq1 3246 . . . 4 (A = B → (A C) = (B C))
43compleqd 3245 . . 3 (A = B → ∼ (A C) = ∼ (B C))
52, 4nineq12d 3242 . 2 (A = B → ( ∼ (C A) ⩃ ∼ (A C)) = ( ∼ (C B) ⩃ ∼ (B C)))
6 df-symdif 3216 . . 3 (CA) = ((C A) ∪ (A C))
7 df-un 3214 . . 3 ((C A) ∪ (A C)) = ( ∼ (C A) ⩃ ∼ (A C))
86, 7eqtri 2373 . 2 (CA) = ( ∼ (C A) ⩃ ∼ (A C))
9 df-symdif 3216 . . 3 (CB) = ((C B) ∪ (B C))
10 df-un 3214 . . 3 ((C B) ∪ (B C)) = ( ∼ (C B) ⩃ ∼ (B C))
119, 10eqtri 2373 . 2 (CB) = ( ∼ (C B) ⩃ ∼ (B C))
125, 8, 113eqtr4g 2410 1 (A = B → (CA) = (CB))
 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-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  df-un 3214  df-dif 3215  df-symdif 3216 This theorem is referenced by:  symdifeq12  3250  symdifeq2i  3252  symdifeq2d  3255
 Copyright terms: Public domain W3C validator