Theorem difeq1i 3323

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

Hypothesis

Ref	Expression
difeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
difeq1i	⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)

Proof of Theorem difeq1i

Step	Hyp	Ref	Expression
1		difeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		difeq1 3320	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)

Syntax hints:   = wceq 1398   ∖ cdif 3198

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-dif 3203

This theorem is referenced by:  difeq12i  3325  indif1  3454  indifcom  3455  difun1  3469  notab  3479  notrab  3486  difprsn1  3817  difprsn2  3818  orddif  4651  resdifcom  5037  resdmdfsn  5062  phplem1  7081  dfn2  9457