Theorem neleq1 2499

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 2292	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	notbid 671	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶))
3		df-nel 2496	. 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)
4		df-nel 2496	. 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1395   ∈ wcel 2200   ∉ wnel 2495

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-in1 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-nel 2496

This theorem is referenced by:  neleq12d  2501  ruALT  4642