Theorem difeq2i 3237

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 3234	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)

Syntax hints:   = wceq 1343   ∖ cdif 3113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-rab 2453  df-dif 3118

This theorem is referenced by:  difeq12i  3238  inssddif  3363  difdif2ss  3379  dif32  3385  difabs  3386  symdif1  3387  notrab  3399  dif0  3479  difdifdirss  3493  dfif3  3533  difpr  3715  dif1o  6406  unfiin  6891