Theorem syl5eleq 2123

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 2113	1 |- (ph -> A e. C)

Syntax hints:   -> wi 4   = wceq 1242   e. wcel 1390

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033

This theorem is referenced by:  syl5eleqr  2124  opth1  3964  opth  3965  eqelsuc  4122  bj-nnelirr  9413