Theorem clelsb3 2142

Description: Substitution applied to an atomic wff (class version of elsb3 1852). (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 1421	. . 3 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
2	1	sbco2 1839	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝐴)
3		nfv 1421	. . . 4 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴
4		eleq1 2100	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	3, 4	sbie 1674	. . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
6	5	sbbii 1648	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴)
7		nfv 1421	. . 3 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
8		eleq1 2100	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
9	7, 8	sbie 1674	. 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
10	2, 6, 9	3bitr3i 199	1 ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 98   ∈ wcel 1393  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036

This theorem is referenced by:  hblem  2145  nfraldya  2358  nfrexdya  2359  cbvreu  2531  sbcel1gv  2821  rmo3  2849  setindel  4263  elirr  4266  en2lp  4278  zfregfr  4298  tfi  4305  bdcriota  10003