Theorem syl6eqelr 2180

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 2094	. 2 |- ( ph -> A = B )
3		syl6eqelr.2	. 2 |- B e. C
4	2, 3	syl6eqel 2179	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082  df-clel 2085

This theorem is referenced by:  eusvnfb  4289  releldm2  5969  mapprc  6423  ixpprc  6490  ixpssmap2g  6498  ixpssmapg  6499  bren  6518  brdomg  6519  mapen  6616  ssenen  6621  ioof  9450  hashfacen  10302  fisum  10839