Theorem eqeltrrid 2284

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 2200	. 2 ⊢ 𝐴 = 𝐵
3		eqeltrrid.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
4	2, 3	eqeltrid 2283	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  dmrnssfld  4930  cnvexg  5208  opabbrex  5970  offval  6147  resfunexgALT  6174  abrexexg  6184  abrexex2g  6186  opabex3d  6187  oprssdmm  6238  unfidisj  6992  residfi  7015  ssfii  7049  djuexb  7119  nqprlu  7633  iccshftr  10088  iccshftl  10090  iccdil  10092  icccntr  10094  mertenslem2  11720  exprmfct  12333  infpnlem1  12555  4sqlem13m  12599  ennnfonelemg  12647  prdsval  12977  prdsbaslemss  12978  grpidvalg  13077  igsumvalx  13093  grppropstrg  13223  releqgg  13428  eqgex  13429  0opn  14350  difopn  14452  tgrest  14513  txbasex  14601  txdis1cn  14622  cnmptid  14625  cnmptc  14626  cnmpt1st  14632  cnmpt2nd  14633  cnmpt2c  14634  hmeoima  14654  hmeocld  14656  fsumcncntop  14911  expcn  14913  plycoeid3  15101