Theorem difeq2 3183

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

Assertion

Ref	Expression
difeq2	⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))

Proof of Theorem difeq2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2201	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	notbid 656	. . 3 ⊢ (𝐴 = 𝐵 → (¬ 𝑥 ∈ 𝐴 ↔ ¬ 𝑥 ∈ 𝐵))
3	2	rabbidv 2670	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐶 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∈ 𝐶 ∣ ¬ 𝑥 ∈ 𝐵})
4		dfdif2 3074	. 2 ⊢ (𝐶 ∖ 𝐴) = {𝑥 ∈ 𝐶 ∣ ¬ 𝑥 ∈ 𝐴}
5		dfdif2 3074	. 2 ⊢ (𝐶 ∖ 𝐵) = {𝑥 ∈ 𝐶 ∣ ¬ 𝑥 ∈ 𝐵}
6	3, 4, 5	3eqtr4g 2195	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ∈ wcel 1480  {crab 2418   ∖ cdif 3063

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 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-rab 2423  df-dif 3068

This theorem is referenced by:  difeq12  3184  difeq2i  3186  difeq2d  3189  disjdif2  3436  ssdifeq0  3440  2oconcl  6329  diffitest  6774  diffifi  6781  undifdc  6805  sbthlem2  6839  isbth  6848  difinfinf  6979  ismkvnex  7022  iscld  12261