Theorem eleqtrrid 2244

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

Hypotheses

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

Assertion

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

Proof of Theorem eleqtrrid

Step	Hyp	Ref	Expression
1		eleqtrrid.1	. 2 |- A e. B
2		eleqtrrid.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2160	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrid 2243	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 2125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-17 1503  ax-ial 1511  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-cleq 2147  df-clel 2150

This theorem is referenced by:  rabsnt  3630  0elnn  4572  tfrexlem  6271  rdgtfr  6311  rdgruledefgg  6312  exmidonfinlem  7107  hashinfom  10629  ennnfonelemhom  12103  exmid1stab  13519