Theorem eleq1a 2211

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 2202	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 159	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135

This theorem is referenced by:  elex22  2701  elex2  2702  reu6  2873  disjne  3416  ssimaex  5482  fnex  5642  f1ocnv2d  5974  tfrlem8  6215  eroprf  6522  ac6sfi  6792  recclnq  7200  prnmaddl  7298  renegcl  8023  nn0ind-raph  9168  iccid  9708  opnneiid  12333  metrest  12675  coseq0negpitopi  12917  bj-nn0suc  13162  bj-inf2vnlem2  13169  bj-nn0sucALT  13176