Theorem eqneltrd 2235

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 2208	. 2 |- ( ph -> ( A e. C <-> B e. C ) )
4	1, 3	mtbird 662	1 |- ( ph -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1331   e. wcel 1480

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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135

This theorem is referenced by:  iseqf1olemnab  10261  ctinfomlemom  11940  nninfalllemn  13202