![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elsb4 | GIF version |
Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
elsb4 | ⊢ ([x / y]z ∈ y ↔ z ∈ x) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1416 | . . . . 5 ⊢ (z ∈ y → ∀w z ∈ y) | |
2 | elequ2 1598 | . . . . 5 ⊢ (w = y → (z ∈ w ↔ z ∈ y)) | |
3 | 1, 2 | sbieh 1670 | . . . 4 ⊢ ([y / w]z ∈ w ↔ z ∈ y) |
4 | 3 | sbbii 1645 | . . 3 ⊢ ([x / y][y / w]z ∈ w ↔ [x / y]z ∈ y) |
5 | ax-17 1416 | . . . 4 ⊢ (z ∈ w → ∀y z ∈ w) | |
6 | 5 | sbco2h 1835 | . . 3 ⊢ ([x / y][y / w]z ∈ w ↔ [x / w]z ∈ w) |
7 | 4, 6 | bitr3i 175 | . 2 ⊢ ([x / y]z ∈ y ↔ [x / w]z ∈ w) |
8 | equsb1 1665 | . . . 4 ⊢ [x / w]w = x | |
9 | elequ2 1598 | . . . . 5 ⊢ (w = x → (z ∈ w ↔ z ∈ x)) | |
10 | 9 | sbimi 1644 | . . . 4 ⊢ ([x / w]w = x → [x / w](z ∈ w ↔ z ∈ x)) |
11 | 8, 10 | ax-mp 7 | . . 3 ⊢ [x / w](z ∈ w ↔ z ∈ x) |
12 | sbbi 1830 | . . 3 ⊢ ([x / w](z ∈ w ↔ z ∈ x) ↔ ([x / w]z ∈ w ↔ [x / w]z ∈ x)) | |
13 | 11, 12 | mpbi 133 | . 2 ⊢ ([x / w]z ∈ w ↔ [x / w]z ∈ x) |
14 | ax-17 1416 | . . 3 ⊢ (z ∈ x → ∀w z ∈ x) | |
15 | 14 | sbh 1656 | . 2 ⊢ ([x / w]z ∈ x ↔ z ∈ x) |
16 | 7, 13, 15 | 3bitri 195 | 1 ⊢ ([x / y]z ∈ y ↔ z ∈ x) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 [wsb 1642 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 |
This theorem is referenced by: peano2 4261 |
Copyright terms: Public domain | W3C validator |