Theorem eleq1a 2159

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 2150	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 158	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1289   ∈ wcel 1438

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084

This theorem is referenced by:  elex22  2634  elex2  2635  reu6  2804  disjne  3336  ssimaex  5365  fnex  5519  f1ocnv2d  5848  tfrlem8  6083  eroprf  6385  ac6sfi  6614  recclnq  6951  prnmaddl  7049  renegcl  7743  nn0ind-raph  8863  iccid  9343  bj-nn0suc  11859  bj-inf2vnlem2  11866  bj-nn0sucALT  11873