Theorem neleq2 1200

Description: Equality theorem for negated membership.

Assertion

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

Proof of Theorem neleq2

Step	Hyp	Ref	Expression
1		eleq2 1150	. . 3 ⊢ (A = B → (C ∈ A ↔ C ∈ B))
2	1	negbid 463	. 2 ⊢ (A = B → (¬ C ∈ A ↔ ¬ C ∈ B))
3		df-nel 1193	. 2 ⊢ (C ∉ A ↔ ¬ C ∈ A)
4		df-nel 1193	. 2 ⊢ (C ∉ B ↔ ¬ C ∈ B)
5	2, 3, 4	3bitr4g 428	1 ⊢ (A = B → (C ∉ A ↔ C ∉ B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092   ∉ wnel 1191

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-nel 1193

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