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  5875  f1ocnv2d  6226  mpoexw  6377  tfrlem8  6483  eroprf  6796  ac6sfi  7086  recclnq  7611  prnmaddl  7709  mpomulf  8168  renegcl  8439  nn0ind-raph  9596  iccid  10159  4sqlem1  12960  4sqlem4  12964  4sqlem11  12973  lssvneln0  14386  lss1d  14396  lspsn  14429  rnglidlmmgm  14509  opnneiid  14887  metrest  15229  coseq0negpitopi  15559  bj-nn0suc  16559  bj-inf2vnlem2  16566  bj-nn0sucALT  16573