Theorem difeq2 3247

Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
difeq2	⊢ (A = B → (C ∖ A) = (C ∖ B))

Proof of Theorem difeq2

Step	Hyp	Ref	Expression
1		compleq 3243	. . . 4 ⊢ (A = B → ∼ A = ∼ B)
2	1	nineq2d 3241	. . 3 ⊢ (A = B → (C ⩃ ∼ A) = (C ⩃ ∼ B))
3	2	compleqd 3245	. 2 ⊢ (A = B → ∼ (C ⩃ ∼ A) = ∼ (C ⩃ ∼ B))
4		df-dif 3215	. . 3 ⊢ (C ∖ A) = (C ∩ ∼ A)
5		df-in 3213	. . 3 ⊢ (C ∩ ∼ A) = ∼ (C ⩃ ∼ A)
6	4, 5	eqtri 2373	. 2 ⊢ (C ∖ A) = ∼ (C ⩃ ∼ A)
7		df-dif 3215	. . 3 ⊢ (C ∖ B) = (C ∩ ∼ B)
8		df-in 3213	. . 3 ⊢ (C ∩ ∼ B) = ∼ (C ⩃ ∼ B)
9	7, 8	eqtri 2373	. 2 ⊢ (C ∖ B) = ∼ (C ⩃ ∼ B)
10	3, 6, 9	3eqtr4g 2410	1 ⊢ (A = B → (C ∖ A) = (C ∖ B))

Syntax hints:   → wi 4   = wceq 1642   ⩃ cnin 3204   ∼ ccompl 3205   ∖ cdif 3206   ∩ 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  df-dif 3215

This theorem is referenced by:  symdifeq1  3248  symdifeq2  3249  difeq12  3380  difeq2i  3382  difeq2d  3385  ssdifeq0  3632