Theorem difeq2i 3382

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 3247	. 2 ⊢ (A = B → (C ∖ A) = (C ∖ B))
3	1, 2	ax-mp 5	1 ⊢ (C ∖ A) = (C ∖ B)

Syntax hints:   = wceq 1642   ∖ cdif 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215

This theorem is referenced by:  difeq12i  3383  dfun3  3493  dfin3  3494  dfin4  3495  invdif  3496  indif  3497  difundi  3507  difindi  3509  dif32  3517  difabs  3518  symdif1  3519  notrab  3532  dif0  3620  undifv  3624  difdifdir  3637  dfif3  3672  cnvin  5035