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Mirrors > Home > ILE Home > Th. List > nfsb2or | GIF version |
Description: Bound-variable hypothesis builder for substitution. Similar to hbsb2 1714 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 1711 | . 2 ⊢ (∀x x = y ∨ ∀x([y / x]φ → ∀x(x = y → φ))) | |
2 | sb2 1647 | . . . . . . 7 ⊢ (∀x(x = y → φ) → [y / x]φ) | |
3 | 2 | a5i 1432 | . . . . . 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 1341 | . . . 4 ⊢ (∀x([y / x]φ → ∀x(x = y → φ)) → ∀x([y / x]φ → ∀x[y / x]φ)) |
6 | df-nf 1347 | . . . 4 ⊢ (Ⅎx[y / x]φ ↔ ∀x([y / x]φ → ∀x[y / x]φ)) | |
7 | 5, 6 | sylibr 137 | . . 3 ⊢ (∀x([y / x]φ → ∀x(x = y → φ)) → Ⅎx[y / x]φ) |
8 | 7 | orim2i 677 | . 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 628 ∀wal 1240 Ⅎwnf 1346 [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-4 1397 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: sbequi 1717 |
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