Theorem eleq1a 2242

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 2233	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 159	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166

This theorem is referenced by:  elex22  2745  elex2  2746  reu6  2919  disjne  3468  ssimaex  5557  fnex  5718  f1ocnv2d  6053  mpoexw  6192  tfrlem8  6297  eroprf  6606  ac6sfi  6876  recclnq  7354  prnmaddl  7452  renegcl  8180  nn0ind-raph  9329  iccid  9882  4sqlem1  12340  4sqlem4  12344  opnneiid  12958  metrest  13300  coseq0negpitopi  13551  bj-nn0suc  13999  bj-inf2vnlem2  14006  bj-nn0sucALT  14013