Theorem difeq1 3274

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 2755	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶} = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶})
2		dfdif2 3165	. 2 ⊢ (𝐴 ∖ 𝐶) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶}
3		dfdif2 3165	. 2 ⊢ (𝐵 ∖ 𝐶) = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶}
4	1, 2, 3	3eqtr4g 2254	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1364   ∈ wcel 2167  {crab 2479   ∖ cdif 3154

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-dif 3159

This theorem is referenced by:  difeq12  3276  difeq1i  3277  difeq1d  3280  uneqdifeqim  3536  diffitest  6948