Theorem eleq1a 2301

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 2292	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 160	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  elex22  2815  elex2  2816  reu6  2992  disjne  3545  ssimaex  5694  fnex  5860  f1ocnv2d  6208  mpoexw  6357  tfrlem8  6462  eroprf  6773  ac6sfi  7056  recclnq  7575  prnmaddl  7673  mpomulf  8132  renegcl  8403  nn0ind-raph  9560  iccid  10117  4sqlem1  12906  4sqlem4  12910  4sqlem11  12919  lssvneln0  14331  lss1d  14341  lspsn  14374  rnglidlmmgm  14454  opnneiid  14832  metrest  15174  coseq0negpitopi  15504  bj-nn0suc  16285  bj-inf2vnlem2  16292  bj-nn0sucALT  16299