Theorem neleq1 2354

Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)

Assertion

Ref	Expression
neleq1	⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))

Proof of Theorem neleq1

Step	Hyp	Ref	Expression
1		eleq1 2150	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	notbid 627	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶))
3		df-nel 2351	. 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)
4		df-nel 2351	. 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
5	2, 3, 4	3bitr4g 221	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103   = wceq 1289   ∈ wcel 1438   ∉ wnel 2350

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-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084  df-nel 2351

This theorem is referenced by:  neleq12d  2356  ruALT  4357