Theorem eqeltrrid 2265

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 2181	. 2 ⊢ 𝐴 = 𝐵
3		eqeltrrid.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
4	2, 3	eqeltrid 2264	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  dmrnssfld  4890  cnvexg  5166  opabbrex  5918  offval  6089  resfunexgALT  6108  abrexexg  6118  abrexex2g  6120  opabex3d  6121  oprssdmm  6171  unfidisj  6920  ssfii  6972  djuexb  7042  nqprlu  7545  iccshftr  9993  iccshftl  9995  iccdil  9997  icccntr  9999  mertenslem2  11543  exprmfct  12137  infpnlem1  12356  ennnfonelemg  12403  grpidvalg  12791  grppropstrg  12894  releqgg  13078  0opn  13476  difopn  13578  tgrest  13639  txbasex  13727  txdis1cn  13748  cnmptid  13751  cnmptc  13752  cnmpt1st  13758  cnmpt2nd  13759  cnmpt2c  13760  hmeoima  13780  hmeocld  13782  fsumcncntop  14026