Theorem eqeltrrid 2227

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 2143	. 2 ⊢ 𝐴 = 𝐵
3		eqeltrrid.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
4	2, 3	eqeltrid 2226	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135

This theorem is referenced by:  dmrnssfld  4802  cnvexg  5076  opabbrex  5815  offval  5989  resfunexgALT  6008  abrexexg  6016  abrexex2g  6018  opabex3d  6019  oprssdmm  6069  unfidisj  6810  ssfii  6862  djuexb  6929  nqprlu  7355  iccshftr  9777  iccshftl  9779  iccdil  9781  icccntr  9783  mertenslem2  11305  exprmfct  11818  ennnfonelemg  11916  0opn  12173  difopn  12277  tgrest  12338  txbasex  12426  txdis1cn  12447  cnmptid  12450  cnmptc  12451  cnmpt1st  12457  cnmpt2nd  12458  cnmpt2c  12459  hmeoima  12479  hmeocld  12481  fsumcncntop  12725