Theorem nineq2 3236

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

Assertion

Ref	Expression
nineq2	⊢ (A = B → (C ⩃ A) = (C ⩃ B))

Proof of Theorem nineq2

Step	Hyp	Ref	Expression
1		nineq1 3235	. 2 ⊢ (A = B → (A ⩃ C) = (B ⩃ C))
2		nincom 3227	. 2 ⊢ (A ⩃ C) = (C ⩃ A)
3		nincom 3227	. 2 ⊢ (B ⩃ C) = (C ⩃ B)
4	1, 2, 3	3eqtr3g 2408	1 ⊢ (A = B → (C ⩃ A) = (C ⩃ B))

Syntax hints:   → wi 4   = wceq 1642   ⩃ 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-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

This theorem is referenced by:  nineq12  3237  nineq2i  3239  nineq2d  3242  ninexg  4098