Theorem compleq 3244

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

Assertion

Ref	Expression
compleq	⊢ (A = B → ∼ A = ∼ B)

Proof of Theorem compleq

Step	Hyp	Ref	Expression
1		nineq12 3237	. . 3 ⊢ ((A = B ∧ A = B) → (A ⩃ A) = (B ⩃ B))
2	1	anidms 626	. 2 ⊢ (A = B → (A ⩃ A) = (B ⩃ B))
3		df-compl 3213	. 2 ⊢ ∼ A = (A ⩃ A)
4		df-compl 3213	. 2 ⊢ ∼ B = (B ⩃ B)
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → ∼ A = ∼ B)

Syntax hints:   → wi 4   = wceq 1642   ⩃ cnin 3205   ∼ ccompl 3206

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

This theorem is referenced by:  compleqi  3245  compleqd  3246  difeq2  3248  compleqb  3544  elsuci  4415  nnsucelr  4429  ncfinlower  4484  sfindbl  4531  el2c  6192  nmembers1lem3  6271