Theorem eleq1a 2310

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 2301	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 160	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is used by:  elex22  2837  elex2  2838  reu6  3015  disjne  3578  ssimaex  5767  fnex  5940  f1ocnv2d  6297  f1o3d  6301  mpoexw  6453  tfrlem8  6594  eroprf  6907  ac6sfi  7208  recclnq  7765  prnmaddl  7863  mpomulf  8322  renegcl  8594  nn0ind-raph  9773  iccid  10343  4sqlem1  13195  4sqlem4  13199  4sqlem11  13208  lssvneln0  14818  lss1d  14828  lspsn  14861  rnglidlmmgm  14941  opnneiid  15407  metrest  15749  coseq0negpitopi  16080  ppiublem1  16308  bj-nn0suc  17212  bj-inf2vnlem2  17219  bj-nn0sucALT  17226