Theorem eqneltrrd 2267

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
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.2	. 2 |- ( ph -> -. A e. C )
2		eqneltrrd.1	. . 3 |- ( ph -> A = B )
3	2	eleq1d 2239	. 2 |- ( ph -> ( A e. C <-> B e. C ) )
4	1, 3	mtbid 667	1 |- ( ph -> -. B e. C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1348   e. wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166

This theorem is referenced by:  ctinf  12385