Theorem eleq1a 2238

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 2229	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 159	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161

This theorem is referenced by:  elex22  2741  elex2  2742  reu6  2915  disjne  3462  ssimaex  5547  fnex  5707  f1ocnv2d  6042  mpoexw  6181  tfrlem8  6286  eroprf  6594  ac6sfi  6864  recclnq  7333  prnmaddl  7431  renegcl  8159  nn0ind-raph  9308  iccid  9861  4sqlem1  12318  4sqlem4  12322  opnneiid  12804  metrest  13146  coseq0negpitopi  13397  bj-nn0suc  13846  bj-inf2vnlem2  13853  bj-nn0sucALT  13860