Theorem eleq1a 2304

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 2295	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 160	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228

This theorem is referenced by:  elex22  2829  elex2  2830  reu6  3006  disjne  3562  ssimaex  5738  fnex  5906  f1ocnv2d  6259  mpoexw  6409  tfrlem8  6549  eroprf  6862  ac6sfi  7155  recclnq  7707  prnmaddl  7805  mpomulf  8264  renegcl  8534  nn0ind-raph  9695  iccid  10258  4sqlem1  13086  4sqlem4  13090  4sqlem11  13099  lssvneln0  14521  lss1d  14531  lspsn  14564  rnglidlmmgm  14644  opnneiid  15029  metrest  15371  coseq0negpitopi  15701  bj-nn0suc  16734  bj-inf2vnlem2  16741  bj-nn0sucALT  16748