Theorem eqeltrrid 2320

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 2236	. 2 |- A = B
3		eqeltrrid.2	. 2 |- ( ph -> B e. C )
4	2, 3	eqeltrid 2319	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203

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 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228

This theorem is referenced by:  dmrnssfld  5020  cnvexg  5300  opabbrex  6097  offval  6274  resfunexgALT  6301  abrexexg  6311  abrexex2g  6313  opabex3d  6314  oprssdmm  6365  unfidisj  7182  residfi  7207  ssfii  7261  djuexb  7335  nqprlu  7862  iccshftr  10327  iccshftl  10329  iccdil  10331  icccntr  10333  mertenslem2  12222  exprmfct  12835  infpnlem1  13057  4sqlem13m  13101  ennnfonelemg  13154  prdsval  13486  prdsbaslemss  13487  grpidvalg  13586  igsumvalx  13602  grppropstrg  13732  releqgg  13937  eqgex  13938  0opn  14871  difopn  14973  tgrest  15034  txbasex  15122  txdis1cn  15143  cnmptid  15146  cnmptc  15147  cnmpt1st  15153  cnmpt2nd  15154  cnmpt2c  15155  hmeoima  15175  hmeocld  15177  fsumcncntop  15432  expcn  15434  plycoeid3  15622