Theorem neleq1 2435

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 2229	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	notbid 657	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶))
3		df-nel 2432	. 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)
4		df-nel 2432	. 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
5	2, 3, 4	3bitr4g 222	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1343   ∈ wcel 2136   ∉ wnel 2431

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 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161  df-nel 2432

This theorem is referenced by:  neleq12d  2437  ruALT  4528