Theorem difeq2d 3323

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 3317	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))

Syntax hints:   → wi 4   = wceq 1395   ∖ cdif 3195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-rab 2517  df-dif 3200

This theorem is referenced by:  difeq12d  3324  exmid1stab  4296  phplem3  7035  phplem4  7036  phplem3g  7037  phplem4dom  7043  phplem4on  7049  fidifsnen  7052  xpfi  7117  sbthlem2  7148  sbthlemi3  7149  isbth  7157  ismkvnex  7345  setsvalg  13102  setsvala  13103