Theorem nineq1 3235

Description: Equality law for anti-intersection. (Contributed by SF, 11-Jan-2015.)

Assertion

Ref	Expression
nineq1	|- (A = B -> (A &ncap C) = (B &ncap C))

Proof of Theorem nineq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . 4 |- (A = B -> (x e. A <-> x e. B))
2	1	nanbi1d 1301	. . 3 |- (A = B -> ((x e. A -/\ x e. C) <-> (x e. B -/\ x e. C)))
3	2	abbidv 2468	. 2 |- (A = B -> {x | (x e. A -/\ x e. C)} = {x | (x e. B -/\ x e. C)})
4		df-nin 3212	. 2 |- (A &ncap C) = {x | (x e. A -/\ x e. C)}
5		df-nin 3212	. 2 |- (B &ncap C) = {x | (x e. B -/\ x e. C)}
6	3, 4, 5	3eqtr4g 2410	1 |- (A = B -> (A &ncap C) = (B &ncap C))

Syntax hints:   -> wi 4   -/\ wnan 1287   = wceq 1642   e. wcel 1710  {cab 2339   &ncap cnin 3205

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 3212

This theorem is referenced by:  nineq2  3236  nineq12  3237  nineq1i  3238  nineq1d  3241  difeq1  3247  ninexg  4098