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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 1419 | . . . . 5 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦) | |
2 | elequ2 1601 | . . . . 5 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦)) | |
3 | 1, 2 | sbieh 1673 | . . . 4 ⊢ ([𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) |
4 | 3 | sbbii 1648 | . . 3 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑥 / 𝑦]𝑧 ∈ 𝑦) |
5 | ax-17 1419 | . . . 4 ⊢ (𝑧 ∈ 𝑤 → ∀𝑦 𝑧 ∈ 𝑤) | |
6 | 5 | sbco2h 1838 | . . 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 1668 | . . . 4 ⊢ [𝑥 / 𝑤]𝑤 = 𝑥 | |
9 | elequ2 1601 | . . . . 5 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥)) | |
10 | 9 | sbimi 1647 | . . . 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 1833 | . . 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 1419 | . . 3 ⊢ (𝑧 ∈ 𝑥 → ∀𝑤 𝑧 ∈ 𝑥) | |
15 | 14 | sbh 1659 | . 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 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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: peano2 4318 |
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