Theorem neleq2 2440

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 2234	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
2	1	notbid 662	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐶 ∈ 𝐴 ↔ ¬ 𝐶 ∈ 𝐵))
3		df-nel 2436	. 2 ⊢ (𝐶 ∉ 𝐴 ↔ ¬ 𝐶 ∈ 𝐴)
4		df-nel 2436	. 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 1348   ∈ wcel 2141   ∉ wnel 2435

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-nel 2436

This theorem is referenced by:  neleq12d  2441