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  4926  cnvexg  5204  opabbrex  5963  offval  6140  resfunexgALT  6162  abrexexg  6172  abrexex2g  6174  opabex3d  6175  oprssdmm  6226  unfidisj  6980  residfi  7001  ssfii  7035  djuexb  7105  nqprlu  7609  iccshftr  10063  iccshftl  10065  iccdil  10067  icccntr  10069  mertenslem2  11682  exprmfct  12279  infpnlem1  12500  4sqlem13m  12544  ennnfonelemg  12563  grpidvalg  12959  igsumvalx  12975  grppropstrg  13094  releqgg  13293  eqgex  13294  0opn  14185  difopn  14287  tgrest  14348  txbasex  14436  txdis1cn  14457  cnmptid  14460  cnmptc  14461  cnmpt1st  14467  cnmpt2nd  14468  cnmpt2c  14469  hmeoima  14489  hmeocld  14491  fsumcncntop  14746  expcn  14748