Theorem syl6eqelr 2129

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 2045	. 2 |- ( ph -> A = B )
3		syl6eqelr.2	. 2 |- B e. C
4	2, 3	syl6eqel 2128	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  eusvnfb  4186  releldm2  5811  bren  6228  brdomg  6229  ioof  8840