Theorem eleq1a 2212

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 2203	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 159	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136

This theorem is referenced by:  elex22  2704  elex2  2705  reu6  2877  disjne  3421  ssimaex  5490  fnex  5650  f1ocnv2d  5982  tfrlem8  6223  eroprf  6530  ac6sfi  6800  recclnq  7224  prnmaddl  7322  renegcl  8047  nn0ind-raph  9192  iccid  9738  opnneiid  12372  metrest  12714  coseq0negpitopi  12965  bj-nn0suc  13333  bj-inf2vnlem2  13340  bj-nn0sucALT  13347