Theorem eqeltrrid 2322

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 2238	. 2 |- A = B
3		eqeltrrid.2	. 2 |- ( ph -> B e. C )
4	2, 3	eqeltrid 2321	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-clel 2230

This theorem is referenced by:  dmrnssfld  5025  cnvexg  5305  opabbrex  6105  offval  6283  resfunexgALT  6310  abrexexg  6320  abrexex2g  6322  opabex3d  6323  oprssdmm  6378  unfidisj  7195  residfi  7220  ssfii  7274  djuexb  7348  nqprlu  7878  iccshftr  10346  iccshftl  10348  iccdil  10350  icccntr  10352  mertenslem2  12247  exprmfct  12860  infpnlem1  13082  4sqlem13m  13126  ballotfilemfrcn0  13217  ennnfonelemg  13238  grpidvalg  13636  igsumvalx  13652  grppropstrg  13774  releqgg  13973  eqgex  13974  prdsval  14115  prdsbaslemss  14116  aprprop  14539  0opn  14997  difopn  15099  tgrest  15160  txbasex  15248  txdis1cn  15269  cnmptid  15272  cnmptc  15273  cnmpt1st  15279  cnmpt2nd  15280  cnmpt2c  15281  hmeoima  15301  hmeocld  15303  fsumcncntop  15558  expcn  15560  plycoeid3  15748