Theorem neleq2 2434

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

Assertion

Ref	Expression
neleq2	⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

Proof of Theorem neleq2

Step	Hyp	Ref	Expression
1		eleq2 2228	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
2	1	notbid 657	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐶 ∈ 𝐴 ↔ ¬ 𝐶 ∈ 𝐵))
3		df-nel 2430	. 2 ⊢ (𝐶 ∉ 𝐴 ↔ ¬ 𝐶 ∈ 𝐴)
4		df-nel 2430	. 2 ⊢ (𝐶 ∉ 𝐵 ↔ ¬ 𝐶 ∈ 𝐵)
5	2, 3, 4	3bitr4g 222	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1342   ∈ wcel 2135   ∉ wnel 2429

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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-cleq 2157  df-clel 2160  df-nel 2430

This theorem is referenced by:  neleq12d  2435