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  2815  elex2  2816  reu6  2992  disjne  3545  ssimaex  5697  fnex  5865  f1ocnv2d  6216  mpoexw  6365  tfrlem8  6470  eroprf  6783  ac6sfi  7068  recclnq  7590  prnmaddl  7688  mpomulf  8147  renegcl  8418  nn0ind-raph  9575  iccid  10133  4sqlem1  12926  4sqlem4  12930  4sqlem11  12939  lssvneln0  14352  lss1d  14362  lspsn  14395  rnglidlmmgm  14475  opnneiid  14853  metrest  15195  coseq0negpitopi  15525  bj-nn0suc  16382  bj-inf2vnlem2  16389  bj-nn0sucALT  16396