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Mirrors > Home > ILE Home > Th. List > hbequid | GIF version |
Description: Bound-variable hypothesis
builder for 𝑥 = 𝑥. This theorem tells us
that any variable, including 𝑥, is effectively not free in
𝑥 =
𝑥, even though 𝑥 is
technically free according to the
traditional definition of free variable.
The proof uses only ax-8 1504 and ax-i12 1507 on top of (the FOL analogue of) modal logic KT. This shows that this can be proved without ax-i9 1530, even though Theorem equid 1701 cannot. A shorter proof using ax-i9 1530 is obtainable from equid 1701 and hbth 1463. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) |
Ref | Expression |
---|---|
hbequid | ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax12or 1508 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))) | |
2 | ax-8 1504 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥)) | |
3 | 2 | pm2.43i 49 | . . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥) |
4 | 3 | alimi 1455 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑥) |
5 | 4 | a1d 22 | . . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)) |
6 | ax-4 1510 | . . . 4 ⊢ (∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)) | |
7 | 5, 6 | jaoi 716 | . . 3 ⊢ ((∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)) |
8 | 5, 7 | jaoi 716 | . 2 ⊢ ((∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)) |
9 | 1, 8 | ax-mp 5 | 1 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 708 ∀wal 1351 = wceq 1353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-gen 1449 ax-8 1504 ax-i12 1507 ax-4 1510 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: equveli 1759 |
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