Theorem difeq2 3248

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 3244	. . . 4 |- (A = B -> ∼ A = ∼ B)
2	1	nineq2d 3242	. . 3 |- (A = B -> (C &ncap ∼ A) = (C &ncap ∼ B))
3	2	compleqd 3246	. 2 |- (A = B -> ∼ (C &ncap ∼ A) = ∼ (C &ncap ∼ B))
4		df-dif 3216	. . 3 |- (C \ A) = (C i^i ∼ A)
5		df-in 3214	. . 3 |- (C i^i ∼ A) = ∼ (C &ncap ∼ A)
6	4, 5	eqtri 2373	. 2 |- (C \ A) = ∼ (C &ncap ∼ A)
7		df-dif 3216	. . 3 |- (C \ B) = (C i^i ∼ B)
8		df-in 3214	. . 3 |- (C i^i ∼ B) = ∼ (C &ncap ∼ B)
9	7, 8	eqtri 2373	. 2 |- (C \ B) = ∼ (C &ncap ∼ B)
10	3, 6, 9	3eqtr4g 2410	1 |- (A = B -> (C \ A) = (C \ B))

Syntax hints:   -> wi 4   = wceq 1642   &ncap cnin 3205   ∼ ccompl 3206   \ cdif 3207   i^i cin 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by:  symdifeq1  3249  symdifeq2  3250  difeq12  3381  difeq2i  3383  difeq2d  3386  ssdifeq0  3633