Theorem eqeltrrid 2284

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

Hypotheses

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

Assertion

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

Proof of Theorem eqeltrrid

Step	Hyp	Ref	Expression
1		eqeltrrid.1	. . 3 |- B = A
2	1	eqcomi 2200	. 2 |- A = B
3		eqeltrrid.2	. 2 |- ( ph -> B e. C )
4	2, 3	eqeltrid 2283	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  dmrnssfld  4929  cnvexg  5207  opabbrex  5966  offval  6143  resfunexgALT  6165  abrexexg  6175  abrexex2g  6177  opabex3d  6178  oprssdmm  6229  unfidisj  6983  residfi  7006  ssfii  7040  djuexb  7110  nqprlu  7614  iccshftr  10069  iccshftl  10071  iccdil  10073  icccntr  10075  mertenslem2  11701  exprmfct  12306  infpnlem1  12528  4sqlem13m  12572  ennnfonelemg  12620  grpidvalg  13016  igsumvalx  13032  grppropstrg  13151  releqgg  13350  eqgex  13351  0opn  14242  difopn  14344  tgrest  14405  txbasex  14493  txdis1cn  14514  cnmptid  14517  cnmptc  14518  cnmpt1st  14524  cnmpt2nd  14525  cnmpt2c  14526  hmeoima  14546  hmeocld  14548  fsumcncntop  14803  expcn  14805  plycoeid3  14993