Theorem syl5eleq 2126

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

Hypotheses

Ref	Expression
syl5eleq.1	|- A e. B
syl5eleq.2	|- ( ph -> B = C )

Assertion

Ref	Expression
syl5eleq	|- ( ph -> A e. C )

Proof of Theorem syl5eleq

Step	Hyp	Ref	Expression
1		syl5eleq.1	. . 3 |- A e. B
2	1	a1i 9	. 2 |- ( ph -> A e. B )
3		syl5eleq.2	. 2 |- ( ph -> B = C )
4	2, 3	eleqtrd 2116	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:  syl5eleqr  2127  opth1  3973  opth  3974  eqelsuc  4156  bj-nnelirr  10078