Theorem difeq2i 3287

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 3284	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)

Syntax hints:   = wceq 1372   ∖ cdif 3162

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 615  ax-in2 616  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-ral 2488  df-rab 2492  df-dif 3167

This theorem is referenced by:  difeq12i  3288  inssddif  3413  difdif2ss  3429  dif32  3435  difabs  3436  symdif1  3437  notrab  3449  dif0  3530  difdifdirss  3544  dfif3  3583  difpr  3774  dif1o  6523  unfiin  7022  m1bits  12213