Theorem neleq2 2520

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 2302	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
2	1	notbid 677	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐶 ∈ 𝐴 ↔ ¬ 𝐶 ∈ 𝐵))
3		df-nel 2516	. 2 ⊢ (𝐶 ∉ 𝐴 ↔ ¬ 𝐶 ∈ 𝐴)
4		df-nel 2516	. 2 ⊢ (𝐶 ∉ 𝐵 ↔ ¬ 𝐶 ∈ 𝐵)
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ∉ wnel 2515

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-nel 2516

This theorem is referenced by:  neleq12d  2521  eupth2lem3lem6fi  16626