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Mirrors > Home > ILE Home > Th. List > nfsb2or | GIF version |
Description: Bound-variable hypothesis builder for substitution. Similar to hbsb2 1765 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.) |
Ref | Expression |
---|---|
nfsb2or | ⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb4or 1762 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | sb2 1698 | . . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | |
3 | 2 | a5i 1481 | . . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑) |
4 | 3 | imim2i 12 | . . . . 5 ⊢ (([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) |
5 | 4 | alimi 1390 | . . . 4 ⊢ (∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) |
6 | df-nf 1396 | . . . 4 ⊢ (Ⅎ𝑥[𝑦 / 𝑥]𝜑 ↔ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) | |
7 | 5, 6 | sylibr 133 | . . 3 ⊢ (∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → Ⅎ𝑥[𝑦 / 𝑥]𝜑) |
8 | 7 | orim2i 714 | . 2 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) → (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥[𝑦 / 𝑥]𝜑)) |
9 | 1, 8 | ax-mp 7 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 665 ∀wal 1288 Ⅎwnf 1395 [wsb 1693 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 |
This theorem depends on definitions: df-bi 116 df-nf 1396 df-sb 1694 |
This theorem is referenced by: sbequi 1768 |
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