Theorem eqneltrd 2558

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.)

Hypotheses

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

Assertion

Ref	Expression
eqneltrd	|- ( ph -> -. A e. C )

Proof of Theorem eqneltrd

Step	Hyp	Ref	Expression
1		eqneltrd.2	. 2 |- ( ph -> -. B e. C )
2		eqneltrd.1	. . 3 |- ( ph -> A = B )
3	2	eleq1d 2519	. 2 |- ( ph -> ( A e. C <-> B e. C ) )
4	1, 3	mtbird 301	1 |- ( ph -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1370   e. wcel 1758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-cleq 2443  df-clel 2446

This theorem is referenced by:  eqneltrrd  2559  omopth2  7120  fpwwe2  8908  sqrneglem  12855  mreexmrid  14680  mplcoe1  17648  mplcoe5  17652  mplcoe2OLD  17654  islln2a  33464  islpln2a  33495  islvol2aN  33539