Theorem neleq1 2607

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 2413	. . 3 ⊢ (A = B → (A ∈ C ↔ B ∈ C))
2	1	notbid 285	. 2 ⊢ (A = B → (¬ A ∈ C ↔ ¬ B ∈ C))
3		df-nel 2519	. 2 ⊢ (A ∉ C ↔ ¬ A ∈ C)
4		df-nel 2519	. 2 ⊢ (B ∉ C ↔ ¬ B ∈ C)
5	2, 3, 4	3bitr4g 279	1 ⊢ (A = B → (A ∉ C ↔ B ∉ C))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710   ∉ wnel 2517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-nel 2519

This theorem is referenced by:  neleq12d  2609