Theorem eqneltrrd 2881

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 2766	. 2 ⊢ (𝜑 → 𝐵 = 𝐴)
3		eqneltrrd.2	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
4	2, 3	eqneltrd 2880	1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144

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

This theorem is used by:  bitsf1  16621  lssvancl2  21226  lbsind2  21361  lindfind2  22129  2atjlej  40629  2atnelvolN  40737  lmod1zrnlvec  49699  veroquaddetzerod  51154