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Mirrors > Home > ILE Home > Th. List > hbim1 | GIF version |
Description: A closed form of hbim 1437. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
hbim1.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hbim1.2 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
Ref | Expression |
---|---|
hbim1 | ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbim1.2 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
2 | 1 | a2i 11 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) |
3 | hbim1.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
4 | 3 | 19.21h 1449 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
5 | 2, 4 | sylibr 137 | 1 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 |
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-5 1336 ax-gen 1338 ax-4 1400 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: nfim1 1463 sbco2d 1840 sbco2vd 1841 |
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