Theorem clelsb3 2888

Description: Substitution applied to an atomic wff (class version of elsb3 2513). (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 1957	. . 3 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
2	1	sbco2 2492	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝐴)
3		nfv 1957	. . . 4 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴
4		eleq1w 2842	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	3, 4	sbie 2484	. . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
6	5	sbbii 2019	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴)
7		nfv 1957	. . 3 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
8		eleq1w 2842	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
9	7, 8	sbie 2484	. 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
10	2, 6, 9	3bitr3i 293	1 ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 198  [wsb 2011   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clel 2774

This theorem is referenced by:  hblem  2891  clelsb3f  2938  cbvreu  3365  sbcel1vOLD  3714  rmo3OLD  3746  iuninc  29941  measiuns  30878  ballotlemodife  31158  bj-nfcf  33493  wl-clelsb3df  33998  ellimcabssub0  40761