![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > clelsb3f | GIF version |
Description: Substitution applied to an atomic wff (class version of elsb3 1952). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) |
Ref | Expression |
---|---|
clelsb3f.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
clelsb3f | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clelsb3f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcri 2276 | . . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴 |
3 | 2 | sbco2 1939 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴) |
4 | nfv 1509 | . . . 4 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴 | |
5 | eleq1w 2201 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
6 | 4, 5 | sbie 1765 | . . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
7 | 6 | sbbii 1739 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴) |
8 | nfv 1509 | . . 3 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴 | |
9 | eleq1w 2201 | . . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
10 | 8, 9 | sbie 1765 | . 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
11 | 3, 7, 10 | 3bitr3i 209 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1481 [wsb 1736 Ⅎwnfc 2269 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-cleq 2133 df-clel 2136 df-nfc 2271 |
This theorem is referenced by: rmo3f 2885 |
Copyright terms: Public domain | W3C validator |