Theorem eqeltrrid 2281

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

Hypotheses

Ref	Expression
eqeltrrid.1	⊢ 𝐵 = 𝐴
eqeltrrid.2	⊢ (𝜑 → 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
eqeltrrid	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem eqeltrrid

Step	Hyp	Ref	Expression
1		eqeltrrid.1	. . 3 ⊢ 𝐵 = 𝐴
2	1	eqcomi 2197	. 2 ⊢ 𝐴 = 𝐵
3		eqeltrrid.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
4	2, 3	eqeltrid 2280	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  dmrnssfld  4925  cnvexg  5203  opabbrex  5962  offval  6138  resfunexgALT  6160  abrexexg  6170  abrexex2g  6172  opabex3d  6173  oprssdmm  6224  unfidisj  6978  residfi  6999  ssfii  7033  djuexb  7103  nqprlu  7607  iccshftr  10060  iccshftl  10062  iccdil  10064  icccntr  10066  mertenslem2  11679  exprmfct  12276  infpnlem1  12497  4sqlem13m  12541  ennnfonelemg  12560  grpidvalg  12956  igsumvalx  12972  grppropstrg  13091  releqgg  13290  eqgex  13291  0opn  14174  difopn  14276  tgrest  14337  txbasex  14425  txdis1cn  14446  cnmptid  14449  cnmptc  14450  cnmpt1st  14456  cnmpt2nd  14457  cnmpt2c  14458  hmeoima  14478  hmeocld  14480  fsumcncntop  14724