Theorem eqneltrd 2174

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 2147	. 2 |- ( ph -> ( A e. C <-> B e. C ) )
4	1, 3	mtbird 630	1 |- ( ph -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1284   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by: (None)