Theorem neleq12d 3044

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 2834	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
4	3	notbid 319	. 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐷))
5		df-nel 3040	. 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)
6		df-nel 3040	. 2 ⊢ (𝐵 ∉ 𝐷 ↔ ¬ 𝐵 ∈ 𝐷)
7	4, 5, 6	3bitr4g 315	1 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119   ∉ wnel 3039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2732  df-clel 2815  df-nel 3040

This theorem is referenced by:  neleq1  3045  neleq2  3046  ru  3728  chnrev  18591  uhgrspan1  29397  nbgrnself  29453  nbgrnself2  29454  finsumvtxdg2size  29644  noinfepregs  35321  fsetsnprcnex  47525  isubgr3stgrlem6  48469  grlimedgnedg  48629