Theorem eqneltrrd 2927

Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 13-Nov-2019.)

Hypotheses

Ref	Expression
eqneltrrd.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqneltrrd.2	⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)

Assertion

Ref	Expression
eqneltrrd	⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)

Proof of Theorem eqneltrrd

Step	Hyp	Ref	Expression
1		eqneltrrd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	eqcomd 2832	. 2 ⊢ (𝜑 → 𝐵 = 𝐴)
3		eqneltrrd.2	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
4	2, 3	eqneltrd 2926	1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1658   ∈ wcel 2166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-cleq 2819  df-clel 2822

This theorem is referenced by:  bitsf1  15542  lssvancl2  19303  lbsind2  19441  lindfind2  20525  2atjlej  35555  2atnelvolN  35663  lmod1zrnlvec  43131