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  5027  cnvexg  5307  opabbrex  6107  offval  6285  resfunexgALT  6312  abrexexg  6322  abrexex2g  6324  opabex3d  6325  oprssdmm  6380  unfidisj  7197  residfi  7222  ssfii  7276  djuexb  7350  nqprlu  7880  iccshftr  10351  iccshftl  10353  iccdil  10355  icccntr  10357  mertenslem2  12253  exprmfct  12866  infpnlem1  13088  4sqlem13m  13132  ballotfilemfrcn0  13223  ennnfonelemg  13244  grpidvalg  13642  gzsumvalx  13658  grppropstrg  13773  releqgg  13972  eqgex  13973  prdsval  14122  prdsbaslemss  14123  aprprop  14546  0opn  15002  difopn  15104  tgrest  15165  txbasex  15253  txdis1cn  15274  cnmptid  15277  cnmptc  15278  cnmpt1st  15284  cnmpt2nd  15285  cnmpt2c  15286  hmeoima  15306  hmeocld  15308  fsumcncntop  15563  expcn  15565  plycoeid3  15753