Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cbvabw | GIF version |
Description: Version of cbvab 2288 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
Ref | Expression |
---|---|
cbvabw.1 | ⊢ Ⅎ𝑦𝜑 |
cbvabw.2 | ⊢ Ⅎ𝑥𝜓 |
cbvabw.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvabw | ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvabw.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfsbv 1934 | . . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
3 | equequ2 1700 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧)) | |
4 | 3 | imbi1d 230 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑧 → 𝜑))) |
5 | 4 | albidv 1811 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
6 | sb6 1873 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
7 | sb6 1873 | . . . . . 6 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)) | |
8 | 5, 6, 7 | 3bitr4g 222 | . . . . 5 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) |
9 | 2, 8 | sbiev 1779 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
10 | cbvabw.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
11 | cbvabw.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
12 | 10, 11 | sbiev 1779 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
13 | 12 | sbbii 1752 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
14 | 9, 13 | bitr3i 185 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
15 | df-clab 2151 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
16 | df-clab 2151 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
17 | 14, 15, 16 | 3bitr4i 211 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}) |
18 | 17 | eqriv 2161 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 = wceq 1342 Ⅎwnf 1447 [wsb 1749 ∈ wcel 2135 {cab 2150 |
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-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 |
This theorem is referenced by: cbvsbcw 2973 |
Copyright terms: Public domain | W3C validator |