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 1398   ∈ 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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2213

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

This theorem is referenced by:  elex22  2819  elex2  2820  reu6  2996  disjne  3550  ssimaex  5716  fnex  5884  f1ocnv2d  6237  mpoexw  6387  tfrlem8  6527  eroprf  6840  ac6sfi  7130  recclnq  7655  prnmaddl  7753  mpomulf  8212  renegcl  8482  nn0ind-raph  9641  iccid  10204  4sqlem1  13024  4sqlem4  13028  4sqlem11  13037  lssvneln0  14452  lss1d  14462  lspsn  14495  rnglidlmmgm  14575  opnneiid  14958  metrest  15300  coseq0negpitopi  15630  bj-nn0suc  16663  bj-inf2vnlem2  16670  bj-nn0sucALT  16677