Theorem difeq1 3083

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 2595	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶} = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶})
2		dfdif2 2981	. 2 ⊢ (𝐴 ∖ 𝐶) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶}
3		dfdif2 2981	. 2 ⊢ (𝐵 ∖ 𝐶) = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶}
4	1, 2, 3	3eqtr4g 2138	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1284   ∈ wcel 1433  {crab 2352   ∖ cdif 2970

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-dif 2975

This theorem is referenced by:  difeq12  3085  difeq1i  3086  difeq1d  3089  uneqdifeqim  3328  diffitest  6371