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  2816  elex2  2817  reu6  2993  disjne  3546  ssimaex  5703  fnex  5871  f1ocnv2d  6222  mpoexw  6373  tfrlem8  6479  eroprf  6792  ac6sfi  7080  recclnq  7602  prnmaddl  7700  mpomulf  8159  renegcl  8430  nn0ind-raph  9587  iccid  10150  4sqlem1  12951  4sqlem4  12955  4sqlem11  12964  lssvneln0  14377  lss1d  14387  lspsn  14420  rnglidlmmgm  14500  opnneiid  14878  metrest  15220  coseq0negpitopi  15550  bj-nn0suc  16495  bj-inf2vnlem2  16502  bj-nn0sucALT  16509