Theorem eleq1a 2265

Description: A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)

Assertion

Ref	Expression
eleq1a	⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Proof of Theorem eleq1a

Step	Hyp	Ref	Expression
1		eleq1 2256	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 160	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

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:  elex22  2775  elex2  2776  reu6  2949  disjne  3500  ssimaex  5618  fnex  5780  f1ocnv2d  6122  mpoexw  6266  tfrlem8  6371  eroprf  6682  ac6sfi  6954  recclnq  7452  prnmaddl  7550  mpomulf  8009  renegcl  8280  nn0ind-raph  9434  iccid  9991  4sqlem1  12526  4sqlem4  12530  4sqlem11  12539  lssvneln0  13869  lss1d  13879  lspsn  13912  rnglidlmmgm  13992  opnneiid  14332  metrest  14674  coseq0negpitopi  14971  bj-nn0suc  15456  bj-inf2vnlem2  15463  bj-nn0sucALT  15470