Theorem difeq1 3318

Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
difeq1	⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))

Proof of Theorem difeq1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq 2794	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶} = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶})
2		dfdif2 3208	. 2 ⊢ (𝐴 ∖ 𝐶) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶}
3		dfdif2 3208	. 2 ⊢ (𝐵 ∖ 𝐶) = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶}
4	1, 2, 3	3eqtr4g 2289	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1397   ∈ wcel 2202  {crab 2514   ∖ cdif 3197

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-dif 3202

This theorem is referenced by:  difeq12  3320  difeq1i  3321  difeq1d  3324  uneqdifeqim  3580  diffitest  7075  fundm2domnop0  11108