Theorem difeq2d 3347

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 3341	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∖ cdif 3217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-rab 2537  df-dif 3222

This theorem is used by:  difeq12d  3348  exmid1stab  4347  phplem3  7161  phplem4  7162  phplem3g  7163  phplem4dom  7169  phplem4on  7175  fidifsnen  7178  xpfi  7245  sbthlem2  7282  sbthlemi3  7283  isbth  7291  ismkvnex  7501  ballotfilemfval  13286  setsvalg  13439  setsvala  13440  wexmiddiffilem  17265  wexmiddifxylem  17267