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Mirrors > Home > ILE Home > Th. List > hbimd | GIF version |
Description: Deduction form of bound-variable hypothesis builder hbim 1538. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.) |
Ref | Expression |
---|---|
hbimd.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hbimd.2 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
hbimd.3 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
Ref | Expression |
---|---|
hbimd | ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbimd.3 | . . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
2 | 1 | imim2d 54 | . . 3 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → ∀𝑥𝜒))) |
3 | ax-4 1503 | . . . . 5 ⊢ (∀𝑥𝜓 → 𝜓) | |
4 | 3 | imim1i 60 | . . . 4 ⊢ ((𝜓 → ∀𝑥𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒)) |
5 | ax-i5r 1528 | . . . 4 ⊢ ((∀𝑥𝜓 → ∀𝑥𝜒) → ∀𝑥(∀𝑥𝜓 → 𝜒)) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ ((𝜓 → ∀𝑥𝜒) → ∀𝑥(∀𝑥𝜓 → 𝜒)) |
7 | 2, 6 | syl6 33 | . 2 ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(∀𝑥𝜓 → 𝜒))) |
8 | hbimd.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
9 | hbimd.2 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
10 | 9 | imim1d 75 | . . 3 ⊢ (𝜑 → ((∀𝑥𝜓 → 𝜒) → (𝜓 → 𝜒))) |
11 | 8, 10 | alimdh 1460 | . 2 ⊢ (𝜑 → (∀𝑥(∀𝑥𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))) |
12 | 7, 11 | syld 45 | 1 ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-5 1440 ax-gen 1442 ax-4 1503 ax-i5r 1528 |
This theorem is referenced by: hbbid 1568 19.21ht 1574 equveli 1752 dvelimfALT2 1810 |
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