Theorem clelsb3 2183

Description: Substitution applied to an atomic wff (class version of elsb3 1893). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
clelsb3	⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem clelsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . 3 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
2	1	sbco2 1880	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝐴)
3		nfv 1461	. . . 4 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴
4		eleq1 2141	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	3, 4	sbie 1714	. . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
6	5	sbbii 1688	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴)
7		nfv 1461	. . 3 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
8		eleq1 2141	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
9	7, 8	sbie 1714	. 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
10	2, 6, 9	3bitr3i 208	1 ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 103   ∈ wcel 1433  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077

This theorem is referenced by:  hblem  2186  nfraldya  2400  nfrexdya  2401  cbvreu  2575  sbcel1v  2876  rmo3  2905  setindel  4281  elirr  4284  en2lp  4297  zfregfr  4316  tfi  4323  bdcriota  10674