Theorem eleq1a 2268

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 2259	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 160	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  elex22  2778  elex2  2779  reu6  2953  disjne  3505  ssimaex  5625  fnex  5787  f1ocnv2d  6131  mpoexw  6280  tfrlem8  6385  eroprf  6696  ac6sfi  6968  recclnq  7476  prnmaddl  7574  mpomulf  8033  renegcl  8304  nn0ind-raph  9460  iccid  10017  4sqlem1  12582  4sqlem4  12586  4sqlem11  12595  lssvneln0  14005  lss1d  14015  lspsn  14048  rnglidlmmgm  14128  opnneiid  14484  metrest  14826  coseq0negpitopi  15156  bj-nn0suc  15694  bj-inf2vnlem2  15701  bj-nn0sucALT  15708