Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sbc2iegf | GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
sbc2iegf.1 | ⊢ Ⅎ𝑥𝜓 |
sbc2iegf.2 | ⊢ Ⅎ𝑦𝜓 |
sbc2iegf.3 | ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊 |
sbc2iegf.4 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbc2iegf | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 102 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
2 | simpl 102 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊) | |
3 | sbc2iegf.4 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
4 | 3 | adantll 445 | . . . 4 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
5 | nfv 1421 | . . . 4 ⊢ Ⅎ𝑦(𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) | |
6 | sbc2iegf.2 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
7 | 6 | a1i 9 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝜓) |
8 | 2, 4, 5, 7 | sbciedf 2798 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
9 | 8 | adantll 445 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
10 | nfv 1421 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉 | |
11 | sbc2iegf.3 | . . 3 ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊 | |
12 | 10, 11 | nfan 1457 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) |
13 | sbc2iegf.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
14 | 13 | a1i 9 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Ⅎ𝑥𝜓) |
15 | 1, 9, 12, 14 | sbciedf 2798 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 [wsbc 2764 |
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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sbc 2765 |
This theorem is referenced by: sbc2ie 2829 opelopabaf 4010 |
Copyright terms: Public domain | W3C validator |