Theorem syl6eqelr 2170

Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
syl6eqelr.1	|- ( ph -> B = A )
syl6eqelr.2	|- B e. C

Assertion

Ref	Expression
syl6eqelr	|- ( ph -> A e. C )

Proof of Theorem syl6eqelr

Step	Hyp	Ref	Expression
1		syl6eqelr.1	. . 3 |- ( ph -> B = A )
2	1	eqcomd 2086	. 2 |- ( ph -> A = B )
3		syl6eqelr.2	. 2 |- B e. C
4	2, 3	syl6eqel 2169	1 |- ( ph -> A e. C )

Syntax hints:   -> 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-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:  eusvnfb  4204  releldm2  5831  bren  6251  brdomg  6252  ioof  8994