Theorem neleqtrd 2722

Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
neleqtrd.1	|- ( ph -> -. C e. A )
neleqtrd.2	|- ( ph -> A = B )

Assertion

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

Proof of Theorem neleqtrd

Step	Hyp	Ref	Expression
1		neleqtrd.1	. 2 |- ( ph -> -. C e. A )
2		neleqtrd.2	. . 3 |- ( ph -> A = B )
3	2	eleq2d 2687	. 2 |- ( ph -> ( C e. A <-> C e. B ) )
4	1, 3	mtbid 314	1 |- ( ph -> -. C e. B )

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:  neleqtrrd  2723  smoord  7462  r1tskina  9604  ofccat  13708  mreexexlem2d  16305  opptgdim2  25637  dochnel  36682  stoweidlem26  40243  fourierdlem60  40383  fourierdlem61  40384  sge00  40593  sge0sn  40596  sge0split  40626  iundjiunlem  40676