Theorem difeq2i 3383

Description: Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Hypothesis

Ref	Expression
difeq1i.1	⊢ A = B

Assertion

Ref	Expression
difeq2i	⊢ (C ∖ A) = (C ∖ B)

Proof of Theorem difeq2i

Step	Hyp	Ref	Expression
1		difeq1i.1	. 2 ⊢ A = B
2		difeq2 3248	. 2 ⊢ (A = B → (C ∖ A) = (C ∖ B))
3	1, 2	ax-mp 5	1 ⊢ (C ∖ A) = (C ∖ B)

Syntax hints:   = wceq 1642   ∖ cdif 3207

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:  difeq12i  3384  dfun3  3494  dfin3  3495  dfin4  3496  invdif  3497  indif  3498  difundi  3508  difindi  3510  dif32  3518  difabs  3519  symdif1  3520  notrab  3533  dif0  3621  undifv  3625  difdifdir  3638  dfif3  3673  cnvin  5036