Theorem eqneltrd 2285

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 2258	. 2 |- ( ph -> ( A e. C <-> B e. C ) )
4	1, 3	mtbird 674	1 |- ( ph -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   e. wcel 2160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185

This theorem is referenced by:  nnnninfeq  7151  iseqf1olemnab  10514  fprodunsn  11639  ctinfomlemom  12473  aprirr  13592