Theorem neleq12d 3066

Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Hypotheses

Ref	Expression
neleq12d.1	⊢ (𝜑 → 𝐴 = 𝐵)
neleq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
neleq12d	⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))

Proof of Theorem neleq12d

Step	Hyp	Ref	Expression
1		neleq12d.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
2		neleq12d.2	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
3	1, 2	eleq12d 2854	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
4	3	notbid 321	. 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐷))
5		df-nel 3062	. 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)
6		df-nel 3062	. 2 ⊢ (𝐵 ∉ 𝐷 ↔ ¬ 𝐵 ∈ 𝐷)
7	4, 5, 6	3bitr4g 317	1 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144   ∉ wnel 3061

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-clel 2835  df-nel 3062

This theorem is used by:  neleq1  3067  neleq2  3068  ru  3737  chnrev  18806  uhgrspan1  29889  nbgrnself  29945  nbgrnself2  29946  finsumvtxdg2size  30136  noinfepregs  35857  fsetsnprcnex  48218  isubgr3stgrlem6  49162  grlimedgnedg  49322