Theorem eleq1a 2249

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 2240	. 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	biimprcd 160	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  elex22  2754  elex2  2755  reu6  2928  disjne  3478  ssimaex  5579  fnex  5740  f1ocnv2d  6077  mpoexw  6216  tfrlem8  6321  eroprf  6630  ac6sfi  6900  recclnq  7393  prnmaddl  7491  renegcl  8220  nn0ind-raph  9372  iccid  9927  4sqlem1  12388  4sqlem4  12392  lssvneln0  13464  lss1d  13475  lspsn  13507  opnneiid  13749  metrest  14091  coseq0negpitopi  14342  bj-nn0suc  14801  bj-inf2vnlem2  14808  bj-nn0sucALT  14815