Theorem eqneltrrd 2721

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	|- ( ph -> A = B )
eqneltrrd.2	|- ( ph -> -. A e. C )

Assertion

Ref	Expression
eqneltrrd	|- ( ph -> -. B e. C )

Proof of Theorem eqneltrrd

Step	Hyp	Ref	Expression
1		eqneltrrd.1	. . 3 |- ( ph -> A = B )
2	1	eqcomd 2628	. 2 |- ( ph -> B = A )
3		eqneltrrd.2	. 2 |- ( ph -> -. A e. C )
4	2, 3	eqneltrd 2720	1 |- ( ph -> -. B e. C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618

This theorem is referenced by:  bitsf1  15168  lssvancl2  18946  lbsind2  19081  lindfind2  20157  2atjlej  34765  2atnelvolN  34873  lmod1zrnlvec  42283