Theorem difeq2d 3386

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

Hypothesis

Ref	Expression
difeq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
difeq2d	⊢ (φ → (C ∖ A) = (C ∖ B))

Proof of Theorem difeq2d

Step	Hyp	Ref	Expression
1		difeq1d.1	. 2 ⊢ (φ → A = B)
2		difeq2 3248	. 2 ⊢ (A = B → (C ∖ A) = (C ∖ B))
3	1, 2	syl 15	1 ⊢ (φ → (C ∖ A) = (C ∖ B))

Syntax hints:   → wi 4   = 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:  difeq12d  3387  imagekeq  4245  imain  5173