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Mirrors > Home > MPE Home > Th. List > hbim1 | Structured version Visualization version GIF version |
Description: A closed form of hbim 2112. (Contributed by NM, 2-Jun-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 14 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) |
3 | hbim1.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
4 | 3 | 19.21h 2107 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
5 | 2, 4 | sylibr 223 | 1 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nf 1701 |
This theorem is referenced by: hbim 2112 axc14 2360 |
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