Theorem eqeltrrid 2292

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 2208	. 2 ⊢ 𝐴 = 𝐵
3		eqeltrrid.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
4	2, 3	eqeltrid 2291	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1372   ∈ wcel 2175

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-cleq 2197  df-clel 2200

This theorem is referenced by:  dmrnssfld  4940  cnvexg  5219  opabbrex  5988  offval  6165  resfunexgALT  6192  abrexexg  6202  abrexex2g  6204  opabex3d  6205  oprssdmm  6256  unfidisj  7018  residfi  7041  ssfii  7075  djuexb  7145  nqprlu  7659  iccshftr  10115  iccshftl  10117  iccdil  10119  icccntr  10121  mertenslem2  11818  exprmfct  12431  infpnlem1  12653  4sqlem13m  12697  ennnfonelemg  12745  prdsval  13076  prdsbaslemss  13077  grpidvalg  13176  igsumvalx  13192  grppropstrg  13322  releqgg  13527  eqgex  13528  0opn  14449  difopn  14551  tgrest  14612  txbasex  14700  txdis1cn  14721  cnmptid  14724  cnmptc  14725  cnmpt1st  14731  cnmpt2nd  14732  cnmpt2c  14733  hmeoima  14753  hmeocld  14755  fsumcncntop  15010  expcn  15012  plycoeid3  15200