Theorem eleq1a 2306

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 2297	. 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 2205

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 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-clel 2230

This theorem is referenced by:  elex22  2831  elex2  2832  reu6  3009  disjne  3566  ssimaex  5743  fnex  5911  f1ocnv2d  6267  f1o3d  6271  mpoexw  6422  tfrlem8  6562  eroprf  6875  ac6sfi  7168  recclnq  7723  prnmaddl  7821  mpomulf  8280  renegcl  8550  nn0ind-raph  9713  iccid  10277  4sqlem1  13111  4sqlem4  13115  4sqlem11  13124  lssvneln0  14647  lss1d  14657  lspsn  14690  rnglidlmmgm  14770  opnneiid  15155  metrest  15497  coseq0negpitopi  15827  bj-nn0suc  16860  bj-inf2vnlem2  16867  bj-nn0sucALT  16874