Theorem neleq12d 3038

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 2827	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
4	3	notbid 318	. 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐷))
5		df-nel 3034	. 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)
6		df-nel 3034	. 2 ⊢ (𝐵 ∉ 𝐷 ↔ ¬ 𝐵 ∈ 𝐷)
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113   ∉ wnel 3033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2725  df-clel 2808  df-nel 3034

This theorem is referenced by:  neleq1  3039  neleq2  3040  ru  3735  chnrev  18535  uhgrspan1  29283  nbgrnself  29339  nbgrnself2  29340  finsumvtxdg2size  29531  fsetsnprcnex  47179  isubgr3stgrlem6  48095  grlimedgnedg  48255