Theorem eleq1a 2303

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 2294	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 160	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  elex22  2818  elex2  2819  reu6  2995  disjne  3548  ssimaex  5707  fnex  5876  f1ocnv2d  6227  mpoexw  6378  tfrlem8  6484  eroprf  6797  ac6sfi  7087  recclnq  7612  prnmaddl  7710  mpomulf  8169  renegcl  8440  nn0ind-raph  9597  iccid  10160  4sqlem1  12966  4sqlem4  12970  4sqlem11  12979  lssvneln0  14393  lss1d  14403  lspsn  14436  rnglidlmmgm  14516  opnneiid  14894  metrest  15236  coseq0negpitopi  15566  bj-nn0suc  16585  bj-inf2vnlem2  16592  bj-nn0sucALT  16599