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  5708  fnex  5877  f1ocnv2d  6230  mpoexw  6381  tfrlem8  6487  eroprf  6800  ac6sfi  7090  recclnq  7615  prnmaddl  7713  mpomulf  8172  renegcl  8443  nn0ind-raph  9600  iccid  10163  4sqlem1  12982  4sqlem4  12986  4sqlem11  12995  lssvneln0  14409  lss1d  14419  lspsn  14452  rnglidlmmgm  14532  opnneiid  14915  metrest  15257  coseq0negpitopi  15587  bj-nn0suc  16618  bj-inf2vnlem2  16625  bj-nn0sucALT  16632